SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

Size: px
Start display at page:

Download "SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills"

Transcription

1 SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D.

2 Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E) + 5; = E) n (n ) 5; n = - () () + 5 ( ) (( ) ) (6)( 8 ) (6)(-9) E) p q : p = 5, q = - E) w 5w ; w = (5) ( ) () 5() E) ( ) (y ) ; = -, y = ( ) (() ) ( 5) (6 ) (5) 75 7 Practice Problems: Evaluate the given epression for the value(s) of the variable(s). Evaluate + for =. Evaluate y + 7 for = and y =. Evaluate ( + ) (y + 5) for = and y = Solutions: Combining Like Terms like terms: have eactly the same variable factors E. In the first eample above, and are like terms

3 To simplify an algebraic epression:. Remove parenthesis by distributing. Combine like terms (combine coefficients according to their sign; keep the variables and eponents the same.) E. (5y ) (6y ) 0y 6y Distribute 6y 9 Combine like terms Additional Eamples. Simplify the following algebraic epressions: a. + 8 The like terms are and ; and -8 b. ( 7) 6 Distribute. There are no like terms so we are done. c. 5 [7 0] Distribute the - to eliminate the [ ] Combine like terms 7 5 d. 8 5y 9 + y - - y Practice Problems: Simplify the following epressions:. 6 y + 9y 0. 7(9 8) (y ). ( + 0) + ( + ) Solutions. 0 y 0. 6 y

4 Solving Linear Equations The steps to solve a linear equation in one variable can be summarized as follows:. Simplify the algebraic epression on each side by distributing (if parenthesis are present) and combining like terms.. Move all of the terms containing variables to one side and all of the constant terms (numbers) to the other side. This is accomplished by adding or subtracting the same amount to each side of the equation.. Isolate the variable (usually accomplished by division or multiplication).. Check the solution in the original equation. Eamples: a. 5 = Add 5 to each side = 6 Divide by = To check, substitute into the original: () 5 = 6 5 = = b. 5 No grouping symbols are present, no like terms to combine on either side. Start with step : subtract from each side of the equation = -5-5 = 8 Subtract 5 from each side of the equation. To check: substitute = 8 into the original equation: 5 (8) 5 (8) + 5 = = 9 The solution is correct because the same amount was obtained on both sides of the equation.

5 c. ( ) ( ) Begin by distributing and combining like terms on each side of the equation. + = - - Add to each side = - Subtract from each side. - - = - Divide by = - Check: ( ) ( ) ( ) ( ) ( ) (0) - + = - - = - Practice Problems: Solve the following equations for.. - = ( 8). ( + ) = ( 6). 5( ) = Solutions All real numbers Linear Equations with Fractions Equations are usually easier to solve if they do not contain fractions. To eliminate fractions from an equation, multiply both sides by the LCD (of all epressions). If you choose the correct LCD, no fractions will be present after you have multiplied. Eamples: a. n Multiply both sides (every term) by 5 (the LCD). 5 5 ( 5 ) + 5n = 5( ) + 5n = 5 Subtract from each side n = -7 Divide both sides by

6 n = 7 5 b. 7 Multiply both sides by (the LCD). 7 7 ( ) ( ) Note that needs to be distributed on each side Combine like terms 7 9 Subtract 7 from each side = 9 Divide by 7 7 = 7 Practice Problems: Solve the following equations:. 5 =. + 8 = = Solutions. 5/

7 Solving Inequalities A linear inequality in can be written as a + b < 0. The inequality sign might be any of the following: < less than > greater than less than or equal to greater than or equal to To solve a linear inequality, we can apply any of the procedures that we have already discussed for linear equations with the following eception: When an inequality is multiplied or divided by a negative number, the direction of the inequality sign is reversed. Why? Consider 5 < 0. Multiply (or divide) both sides by -. Is -5 < -0? The relationship between the numbers has changed and the correct relationship is -5 > -0. Eamples: Solve the following inequalities. Use interval notation to epress the solution set. Graph each solution set on a number line. a. + 5 < 7 Subtract 5 < Divide by < 6 b. 5 0 Divide by -5. Remember to reverse the direction of the inequality sign because you are dividing by a negative number!!! 6 c. ( ) 0 Distribute - 8 > + 0 Add -8 > Subtract 0-8 > 7 Divide by 7 - > This should be rearranged so that is read first. This inequality is equivalent to: < - 7

8 d. 6( ) ( ) Note that after subtracting 7 from each side, the variable has dropped out. Decide whether the remaining inequality is true or false. If true, the solution set would be all real numbers. If false, there is no solution set and we write Ø. (This is false so there is no solution.) e. 6 0 Subtract from each side. -6 < 0 Note that again, the variable has dropped out. This time, the resulting inequality is true. Therefore, the solution set is all real numbers. Practice Problems: Solve the following linear inequalities (y + ) < ( y + 5). ( + ) 5. (8 + ) < ( 6 + ) Solutions:.. y 8 5. No solution. All real numbers Translating Words to Algebraic Epressions Key Words Addition: sum, added to, total, more than, greater than, increased by Subtraction: difference, subtracted from, less than, decreased by Multiplication: product, times, multiplied by, twice, of Division: quotient, divided by Remember: Switch order for less than and subtracted from Eamples 8 more than five more than twice a number subtracted from a number twice a number subtracted from ten 7 0 a number decreased by 6 the difference between and

9 the product of eight and a number four times the difference of a number and 8 ( ) 0% of a number twice a number increased by eight the quotient of a number and 5 5 a number divided by three one eighth of a number si less than m m 6 8 three less than four times a number twice the difference of a number and four ( ) Epressions Involving Percents The cost, increased by 8.75% ta: The cost reduced by 0% : Consecutive Integers consecutive integers :, +, + consecutive even integers, +, + consecutive odd integers, +, + Write each of the following as an algebraic equation:. Si less than a number is 6 =. Four times the difference of a number and 9 is 7. ( 9) = 7. Seven more than three times a number is two times the sum of the number and five. 7 + = ( + 5). The sum of a number and the number increased by 7 is 6. + ( + 7) = 6 9

10 5. Si less than three times a number is one-fourth the number. 6 = Practice Problems: Translate the following into algebraic epressions or equations. The product of a number and 8.. The difference of 6 and a number.. The sum of twice a number and 7. One sith of a number 5. Four less than a number 6. 60% of a number 7. Eight less than a number is fifteen 8. The sum of a number and the number decreased by 8 is 9. Nine less than si times a number is three times the difference of the number and seven Solutions = ( - 8) = = ( 7) Solving Application Problems Problems that are presented to us in verbal form can usually be translated into a mathematical equation that can be solved to find the solution to the original problem. Follow the strategy below to help with this process:. Read the problem carefully! It is sometimes helpful to underline key words that indicate a certain mathematical operation.. Define a variable for one of the unknown quantities. Use the statement Let =.. If possible, write epressions for any other unknown quantities using the variable chosen in step.. Write an equation that represents the relationship between the unknown quantities. 5. Solve the equation. Use this solution to answer the question posed by the problem. 6. Check your solution according to the original wording of the problem. Note: If you check only in your equation and you have made an error in forming this equation, you may not notice this error. 0

11 Eamples: a. When seven times a number is decreased by, the result is. What is the number? Let = the number (Let = the unknown quantity.) 7 = (Write an equation.) + + (Solve) 7 = 7 7 = The number is. (Answer the question) Check: Seven times decreased by is ( = ) b. When 0% of a number is added to the number, the result is 5. What is the number? Let = the number (Let = the unknown quantity.) = 5 ***Remember, 0% of a number means 0.0 times the number.0 = = 80 The number is 80. Check: 0% of 80 added to 80 = 5 0.0(80) + 80 = = 5 5 = 5

12 c. One number eceeds another by. The sum of the numbers is 58. What are the numbers? Let = the first number + = second number ( eceeds means is more than: addition) Note that in this problem, there are unknown quantities, therefore a second epression is used to represent the second quantity. + ( + ) = 58 + = = = 7 ( Sum indicates addition.) To find the second number, replace with 7 in the epression +. + = 7 + = The numbers are {7, } Check: 7 + = 58 d. In 00, the price of a sports car was approimately $80,500 with a depreciation of $765 per year. After how many years will the car s value be $6,975? Let = # of years until value is $,0 ( Depreciation means that the car is losing value each year. Be sure to multiply the loss each year (8705) by the number of years ()) = = = 5 After 5 years the car s value will be $0 Check: (5) = = 0 0 = 0

13 e. After a 0% reduction, you purchase a dictionary for $5.0. What was the dictionary s price before the reduction? Let = price before reduction To calculate a percentage reduction, multiply the percentage by the original price, then subtract from the original price. 0.0 = = =.75 The original price was $.75. Check:.75(0.0) = = $5.0 f. The selling price of a scientific calculator is $5. If the markup is 5% of the dealer s cost, what is the dealer s cost of the calculator? Let = dealer s cost To calculate a percentage markup, multiply the percentage by the original price, then add the original price = 5.5 = = The dealer s price is $ Check: (.5) = + = 5 g. The length of a rectangular pool is 6 meters less than twice the width. If the pool s perimeter is 6 meters, what are its dimensions? Let = width 6 = length ***Note less than requires us to subtract from

14 P = l + w 6 = ( 6) + 6 = + 6 = = = The width is The length is () = 6 = 6 6 = 0 Check: () + (0) = = 6 Practice problems: Solve the following problems:. When eight times a number is increased by, the result is 6. Find the number.. When 0% of a number is subtracted from the number, the result is 5. Find the number.. Two numbers differ by eight. The sum of the numbers is. Find the numbers.. After a 0% discount, the price of a new cell phone is $50. Find the price of the phone before the discount. 5. The length of a rectangular field is si feet greater than twice the width. If the perimeter is 8ft, find the length and width of the field. 6. A new pair of sneakers cost $75.5 after ta (rate of 8.75%) has been added. Find the price of the sneakers before ta. (Round your answer to the nearest cent.) Solutions and 5. $50 5. L = 78ft; w = 6 ft 6. $69.8 Rules of Eponents m n mn The Product Rule: b b b When multiplying eponential epressions with the same base, add the eponents.

15 Eamples: a. 5 6 b. 6 6 c. (remember: really has an eponent of that is added to the eponent -) d. ( 5 7 )(9 ) 99 e. 7 ( 5 y)( 6 y ) 0 y m b mn The Quotient Rule: b b 0 n b When dividing eponential epressions with the same base, subtract the eponents. Eamples: a a b b. 0 7 c. 5a b 0 7 7a b The Zero- Eponent Rule: If b is any real number other than 0, b 0 Eamples: a. ( 9) 0 b. () 0 c. 0 (note: the eponent applies only to the, NOT to the ) d. 9 0 (note: the eponent applies only to the 9, not to the -.) The Negative Eponent Rule If b is any real number other than 0 and n is a natural number, then Eamples: b n n b a b. c. ( ) ( ) 9 5

16 m n mn The Power Rule: ( b ) b When an eponential epression is raised to a power, multiply the eponents. Eamples: a. 9 ( ) b. ( 55 ) 5 6 c. ( ) n n n Products to Powers: ( ab) a b When a product is raised to a power, raise each factor to that power. Eamples: a. 8 ( 6 ) 6 ( ) 6 b ( y ) ( ) ( ) ( y ) 7 y Quotients to Powers n a n a If b is a nonzero real number, then ( ) n b b When a quotient is raised to a power, raise the numerator and denominator to that power. Eamples: 6 ( 6) 6 a. ( ) b. y y y ( ) ( ) 8 6 Simplifying Eponential Epressions An epression is simplified when:. No parenthesis appear. No powers are raised to powers. Each base occurs only once. No negative eponents or 0 eponents appear. Eamples: a. 5a b 7 7a b 6 7 5a b 7 5a b (Divide the coefficients, subtract eponents.) b. (0 ) 6 (0 ) 000 third power in the denominator.) (Eliminate the negative eponent, raise each factor to the 6

17 Practice Problems Simplify the following:. 8 y 8 7 y. ( ) (y ). (5 y )( 7 y 0 ). (5y ) 5. ( 5 y ) 0 Solutions.. y 5 7 y 7. 5 y. 5y 6 5. Multiplying Polynomials (Monomial)( Monomial): Multiply coefficients, add eponents of like bases. Eamples: a. ( )( ) 8 b ( y )( 6y ) y (Binomial)(Binomial) We can use the acronym FOIL to describe how to multiply two binomials: E. ( + 6)( 5) F stands for the first terms in each binomial: O stands for the outer terms: 5 5 I stands for the inner terms: 6 6 L stands for the last terms in each binomial: Combine like terms and write your answer in standard form: Eamples: a. ( 5)(7 + ) ( )(7) ()() ( 5)(7) ( 5)() Practice combining the outer and inner terms (if like) mentally and just writing the final answer. It will be important to be able to perform this multiplication quickly as you progress through mathematics courses. b. (7 )( 5)

18 c. (7 5)( ) Note: there are no like terms to combine here. Practice Problems Multiply the following:. ( 6)( + 8). ( 9)( + 5). (y 5)(y + 7). ( + 7)(y 5) Solutions: y + y 5. 8y 0 + 8y 5 Factoring Polynomials Factoring is the process of breaking a polynomial into a product. It is the reverse of multiplying. Common Factors Step of factoring is always to remove the GCF (greatest common factor). Remove means divide each term by the GCF. E. 6 The GCF of 6 and - is 8. Write the GCF outside a set of parenthesis. Mentally divide each term by 8. Write each resulting quotient inside the parenthesis. Ans: 8( ) Note: This answer can be checked by distributing. When the is distributed, the result should match the original problem. If it does not, the factorization is not correct. E. 6 8 The GCF is 6. Note that when variables are involved, use the smallest eponent of each common variable as part of the GCF. 6 ( ) (Remember to subtract eponents when dividing like bases). Practice Problems Factor each of the following:.. 8y y

19 Solutions. ( ). y(y ). ( 5 + ) Factoring Trinomials of the Form a b c with a = Remember, factoring is the reverse of multiplying. Eamine these multiplication problems: a.( )( 5) b.( + 6)( + ) c. ( + )( ) d. ( + 5)( ) Our goal is to get from these answers back to the factored form. Use the Trial and Error method, incorporating some guidelines. E. 7 6 (a =, b = 7, c = 6) Form of answer: ( + )( + ) We must fill in the blanks with factors whose product is c Factors of 6 (c): 6 Which pair adds up to 7 (b): 6 ( + 6)( + ) Check by multiplying (FOIL) Eamples: a. 8 5 (a =, b = 8, c = 5) Form of answer: ( + )( + ) We must fill in the blanks with factors whose product is c Factors of 5 (c): 5 5 Which pair adds up to 8 (b): 5 ( + 5)( + ) Check by multiplying (FOIL) Note that ( + )( + 5) is also a correct answer. The order of the factors in the answer does not matter. b. 5 Factors of 5 that add up to - are -9 and -5 ( 9)( 5) 9

20 c. 5 Factors of -5 that add up to - are -5 and + ( 5)( + ) d. 0 Factors of -0 that add up to + are +5 and - ( + 5)( ) Note: All of these answers can be checked by FOILing!!! ****Notice that when the sign of c (the constant) is +, the signs in each factor of your answer will be the same; the sign of b (the middle term) will tell you if both are or +. ****Notice that when the sign of c (the constant) is -, the signs in each factor of your answer will be different; the sign of b (the middle term) will tell you the sign of the larger number. Practice Problems y + y 0. y 5y y y 8 Solutions:. ( + )( + 9). (y + 5) (y ). (y 8)(y 7). Prime (cannot be factored) 5.(y )(y + ) Factoring Trinomials of the form a b c with a Use trial and error but there will be more possibilities to consider. E. 5 The only way to break up Our possibilities are: is. The only way to break up is. ( + )( + ) OR ( + )( + ) A quick check of the inners and outers (from FOIL) will determine the correct factorization. They must produce 5 when combined. ( + )( + ) OR ( + )( + ) inners:, outers: inners:, outers: 6 combined: 5 combined: 7 The first factorization is correct!! 0

21 Eamples: a. 6 7 The possible factors of 6 are 6 and. The possible factors of are, 6 and (or each pair with both negative factors). The sign of b (-7) is negative indicating that the factors of will both be negative. Trial : ( )( ) outers: - inners: -6 combined: -8 (incorrect) Trial : ( )( ) outers: -9. inners: -8 combined: -7 (correct) b. 9 5 The possible factors of and. 9 are 9 and. The possible factors of - are,, Trial : ( )( + ) outers: inners: - combined: -9 (incorrect) Trial : (9 + )( ) outers -6 inners: combined: -5 (incorrect) Note: After checking the outers, you might quickly decide that it appears too large to continue and go on to the net trial. Trial : (9 - )( + ) outers: 9 inners: - combined: 5 (correct) Practice Problems:. 5. 5y y Solutions:. ( + 5)(( ). (5y )(y 6). ( + )( + ) Factoring the Difference of Two Squares When factoring an epression containing terms separated by a - sign, each of which is a perfect square, the factorization will always be of the following form: A B ( A B)( A B) Note: the factors in your answer can be in either order. It doesn t matter if the + is first or the - is first.

22 Eamples: a. A variable epression containing an even eponent is a perfect square, is also a perfect square. ( + )( ) b. 6 y 9 = (6 + 7y)(6 7y) Practice Problems. 5. y Solutions. ( 5)( + 5). (y 9)(y + 9). (0 - )(0 + ) Repeated Factorizations In order to completely factor an epression, it must be broken down as far as possible. This often requires more than one factoring step. Always remove the GCF first if possible. Eamples: a. 8 First, recognize the epression as the difference of squares (9 )(9 ) Note that the second factor is again the difference of squares and should be factored again. (9 )( )( ) b First, remove (divide by) the GCF of (5 5 6) Continue factoring by trial and error (5 + )(5 ) Practice Problems. 8 9y Solutions:. (9( + y)( y). ( + )( ). ( + )( )

23 Rational Epressions Simplifying a Rational Epression To simplify a rational epression:. Factor the numerator and denominator completely. Divide both the numerator and denominator by any common factors.. State any numbers that are ecluded from the domain of the original epression. If the epression is simplified, the numerator and denominator will not contain any common factors other than. Eamples: Simplify. 8 ( ) a. Factor the top and bottom ( )( ) = The top and bottom were divided by the common factor: (-). The restrictions on the domain were stated. y y 5 ( y 5)( y ) y 5 b. y, (Note: the restrictions include ALL y 5y ( y )( y ) y values that make the original epression undefined, not only those that make the simplified version undefined. Practice Problems: Simplify the following epressions:. + 8 Solutions:. y 7y+0 y y y y+. 5

24 Multiplication and Division of Rational Epressions Multiplying Rational Epressions. Factor all numerators and denominators completely.. Divide numerators and denominators by common factors (you can divide one factor anywhere on the top by the identical factor anywhere on the bottom.). Multiply the remaining factors in the numerator; multiply the remaining factors in the denominator. Eamples: a. Factor ( )( ) ( ) ( )( ) Divide the common factors ( ), ( ) and ( + ), b ( )( ) ( )( 9) ( 9) Dividing Rational Epressions To divide rational epressions, multiply the first epression by the reciprocal of the second epression (as we do with rational numbers). Remember, the reciprocal of an epression is formed by inverting (flipping it upside down). Follow the steps for multiplication of rational epressions. Eclude from the domain, any value(s) that makes the original epressions undefined, as well as any value(s) that makes the inverted epression undefined.

25 Eamples: a. 8 8 Keep the first epression, multiply by the reciprocal of the second. Factor ( )( ) ( ) Divide the common factors ( ), b Keep the first epression, multiply by the reciprocal of the second. Factor ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) Divide the common factors Multiply numerators and denominators. ( ) ( )( ),,,, Practice Problems: Perform the indicated operation and simplify (+) y 8 z y z Solutions: (+) ( ) y 7 8 5

26 6 Addition and Subtraction of Rational Epressions Same Denominator Add or subtract the numerators, the denominator remains the same; simplify. Eamples: a. 6 6 Add the numerators, denominator remains the same 6 6 Factor to simplify. ) )( ( ) )( ( Divide the like factors., b. 6 6 Subtract numerators (remember to distribute the - sign) ) ( Factor to simplify ) ( Different Denominators The epressions must be rewritten as equivalent epressions containing the least common denominator (LCD). To find the LCD:. Factor each denominator completely.. List the factors of the first denominator.. Add to the list, any factors of the second denominator that do not appear in the list (there is no need to repeat any common factors).. The product of all the factors in the list is the LCD of the epressions.

27 Eamples: a. Find the LCD of and 0 ( 5)( ) 6 ( )( ) Factor each denominator completely. Factors of first denominator: ( + 5) ( ). From the second denominator, add ( + ) to the list. So the LCD is ( + 5)( )( + ). Note: these factors can appear in any order but all three must appear in the LCD. 5 b. Find the LCD of and 5 5 Factor each denominator completely. 5 5 (5 )(5 ) Factors of first denominator: (5 + ). From the second denominator, add (5 ) to the list. (There is no need to repeat the identical factor.) So the LCD is (5 + )(5 ). Note: these factors can appear in any order. To add or subtract rational epressions that have different denominators:. Find the LCD of the rational epressions.. Rewrite each epression as an equivalent one using the LCD. (Multiply the numerator and denominator by any factor needed to change the denominator into the LCD.). Add or subtract the numerators; keep the denominator the same.. Simplify if possible. 7

28 Eamples: 8 a. LCD is ( )( + ). To find the equivalent epressions, multiply the first fraction by ( + ) and the second fraction by ( ). Remember, multiply BOTH the numerator and denominator. 8( ) ( ) ( )( ) ( )( ) Add the numerators (remember to distribute!) 8( ) ( ) ( )( ) simplified ( )( ) ( )( ) Factor to see if the epression can be 0( ) ( )( ), There are no common factors to divide. b. 7 6 Factor to find the LCD. ( 6)( ) ( 6)( ) LCD: ( 6)( + )( ) To find the equivalent epressions, multiply the first fraction by ( ) and the second fraction by ( + ). Remember, multiply BOTH the numerator and denominator. ( ) ( ) ( 6)( )( ) ( 6)( )( ) Subtract the numerators (remember to distribute!) ( ) ( ) ( 6)( )( ) ( 6)( )( ) 5 ( 6)( )( ) 5,, Practice Problems: Perform the indicated operation and simplify Solutions: (+). 8 ( )( 5). +8 ( )(+) 5. (+)( 7) 8

29 Solving Rational Equations A rational equation contains one or more rational epressions (remember, these contain variables in the denominator). Follow the same procedure for solving as above with linear equations containing fractions. Sometimes, one or more denominators will need to be factored in order to determine the LCD. Remember to avoid any values that make any rational epression involved undefined (values that make the denominator = 0). Eample: 5 8 LCD: 8 (all denominators divide evenly into 8). Multiply 9 8 both sides by ( ) 8( ) Note that 8 needs to be distributed on each side Add 6 to each side = 7 6 Add 6 to each side = 7 Divide by = Check:

30 Practice Problems: Solve the following equations = 0 Solutions:. 6 + = 5. + = , Comple Fractions (Comple Rational Epressions) Simplifying To simplify a comple rational epression:. Find the LCD of all the rational epressions in the numerator and denominator.. Multiply each term of the epression by the LCD (this should eliminate all of the rational epressions).. Simplify the resulting epression by factoring (if possible) and dividing out common factors. Eamples: a. The LCD is. Multiply each term by. 6 Factor ( ) b. y y The LCD is y. Multiply each term by y. y y y y y y y 0, y 0 (No factoring is possible.) 0

31 Practice Problems: Simplify the following rational epressions: y + y Solutions: (5 ). y y+ Solving Systems of Linear Equations Substitution Method Steps:. Solve one of the equations (either one) for one variable (either one).. Substitute the epression obtained in Step into the other equation. (only one variable should now be present.). Solve this resulting equation.. Re-substitute the answer obtained in Step into the either original equation. Solve for the other variable. 5. Check the solution in BOTH original equations. Eamples: Solve the following systems of equations by substitution. a. y = - () (The equations are numbered so that they y = + 7 () can be easily referred to.) Equation () is already solved for y so step can be omitted. The epression + 7 will be substituted into Equ. () in place of y: y = - ( + 7) = - Distribute and solve for. 6 = - - = - - = 8 = - Re-substitute into equ () (or equ () I just chose equ ()) to find y: y = + 7 y = (-) + 7 y = y = Solution (-, )

32 Check: y = - y = + 7 (-) () = - = (-) = - = = - true = true Note: When both lines are graphed, the point of intersection is (-, ) as can be seen on the graph below: b. + 5y = () - + 6y = 8 () It looks like it will be easiest to solve equ. () for (the coefficient is - which should not create any fractions when solving.) - + 6y = 8 - = 8 6y Divide all terms by - = y = 6y -8 Substitute into equ. (): + 5y = (6y 8) + 5y = y 6 + 5y = 7y 6 = 7y = 7 y =

33 Re-substitute into equ. () (or equ (). Note: As we have already isolated in our first step, we could also re-substitute into that epression instead ( = 6y 8) as long as we are sure that no mistakes have been made in isolating y = () = = 8 - = = - Solution: (-, ) Check: + 5y = - + 6y = 8 (-) + 5() = -(-) + 6() = = + 6 = 8 = true 8 = 8 true c. y () y 7 () Both equations have one variable already solved for. Let s substitute equ. () into equ. (): y 7 Multiply all terms by to clear fractions. + 8 = Solve for 5 = -0 = - Re-substitute into equ. () to find y. y 7 y ( ) 7 y = y = Solution (-, ) Check: y y 7 ( ) ( ) 7 7 = true = true

34 Practice Problems Solve the following systems by substitution:. y =. y = 7. y = 5y = + y = ½ y = Solutions:. (, ). (, ). (/8, -/) Addition Method Some tetbooks refer to this method as solving by elimination, meaning that one variable is eliminated (usually by addition). In this method, the like terms in each equation are lined up, added (so that one variable is eliminated) and the resulting equation is solved. This answer is then substituted into either original equation to find the other value. An eample of the simplest type is provided below with more detailed steps to follow: E. Solve by addition: + y = 6 y = - Add the equations together (notice that the ys drops out). + y = 6 y = - = = Solve for Substitute into either original to find y: + y = 6 + y = 6 y = - The solution is (, -) It can be verified that this solution is correct when it is substituted into BOTH of the original equations. Sometimes adding the original equations will not result in one of the variables being eliminated. We can adjust one (or both) equations to force this to happen. To do this, multiply one (or both) by nonzero constants so that the coefficients of one of the variables are opposites. Then, when the equations are added, that variable will be eliminated.

35 E. + 5y = () - + 6y = 8 () If equ () is multiplied by (every term), the system becomes: + 5y = - + y = 6 Add the resulting equations: 7y = 7 y = Substitute into either original equation (I ll use equ ) + 5() = + 5 = = - = - Solution: (-, ) Check this answer in BOTH original equations to be sure that it is correct. Summary of Steps:. Rewrite (if necessary) each equation in the form A + By = C (this is so like terms, and the = signs are lined up).. If necessary, multiply one or both equations by a nonzero constant (so that the coefficients of one of the variables are opposites).. Add the resulting equations.. Solve for the remaining variable. 5. Substitute into either of the original equations to find the other variable. 6. Check the solutions in BOTH of the original equations. Eamples: Solve the following equations using the addition method. a. 7y = () 6 + 5y = 7 () In this case, we can multiply equ () by - to make the coefficients of opposites. To indicate this in your work, write the following: -( 7y = ) 6 + 5y = y = y = 7 Add the equations 9y = -9 y = - Substitute into equ () (or equ ()) 5

36 7(-) = equ () + 7 = = 6 = Solution: (, -) Check: 7y = 6 + 5y = 7 () 7(-) = 6() + 5(-) = = - 5 = 7 = 7 = 7 a. + y = y = 0 (Multiply the top equation by -5 and the bottom equation by ) -0 5y = y = 60 (Add) -5y = 0 (Divide by -5) y = - Substitute into either original equation (I used the top one): + () = -6 = -6 = - = - The solution is (-, -). Check this in both original equations. Practice Problems: Solve the following systems of equations by the Addition method. y = -. y =. + = -5y -+6y = y = 6 8 = 7 Solutions:. (-6, -). (, ). (8, -5) 6

37 Linear Systems with No Solution or Infinitely Many Solutions It is possible that a linear system has no solution (no ordered pair satisfies BOTH equations at the same time) or infinitely many solutions (there is an endless list of ordered pairs that satisfy both equations). If the equations were graphed on the same set of aes, a system with no solution would be two parallel lines. A system with infinitely many solutions would be the same identical line and any point on that line is a solution. E. Solve the following system by addition: 6 + y = 7 y = Line up in A + By = C form: 6 + y = 7 () + y = () Multiply equ () by y = 7-6 y = - Add = 0 = Both variables were eliminated and the resulting equation is False. Therefore, this system has no solution. We can use the symbol for empty set to indicate this: Ø E. Solve the following system by addition: y = () y = () Multiply equ () by-: y = - + y = - Add = 0 Both variables were eliminated and the resulting equation is True. Therefore, this system has an infinite number of solutions. Set builder notation (using either original equation) is often used to indicate all of the solutions: {(, y) y = } OR {(, y) y = } These are read the set of all ordered pairs (, y) such that y = OR the set of all ordered pairs (, y) such that y = meaning that the equation in the solution must be satisfied by the ordered pair in order to be a solution. Practice Problems Solve the following systems by either method. y = 7. + y = + 8y = - - y = 5 7

38 Solutions:. Infinite number of solutions. No solution Radicals Product Rule for Square Roots (a, b are nonnegative real numbers) ab a b and a b ab The square root of a product is the product of the square roots. Eamples: a b We will use this property to simplify square roots. Simplifying Square Roots To be simplified, the radicand must have no perfect square factors other than. Break the radicand into the product of a perfect square and another number, simplify by removing the square root of the perfect square. Eamples: a is a perfect square that is a factor of 7. Its square root is which is written on the outside of the radical sign. b c Note: the last eample could have also been simplified by the following series of steps: If you do not use the largest perfect square that is a factor at first, just keep simplifying until there are no more perfect square factors. If you remove another square root, multiply it by the number that is already outside the radical sign (when the was removed, it s value was multiplied by the that was already outside). Note: Algebraic epressions with even eponents are perfect squares. 8

39 Eamples: a. b. (because 8 (because ) 8 ) Use this fact (in conjunction with the rules already learned) to simplify the following epressions: a First, the radicands are multiplied; net the radicand is broken into factors using one that is a perfect square, simplify. b Practice Problems: Simplify the following radical epressions: y y. 0y 8 5 Solutions:. 8 y 0. 6y. y 0 9

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

9.3 OPERATIONS WITH RADICALS

9.3 OPERATIONS WITH RADICALS 9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in

More information

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Polynomials and Factoring

Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

More information

SECTION P.5 Factoring Polynomials

SECTION P.5 Factoring Polynomials BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

More information

Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

More information

Five 5. Rational Expressions and Equations C H A P T E R

Five 5. Rational Expressions and Equations C H A P T E R Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

The majority of college students hold credit cards. According to the Nellie May

The majority of college students hold credit cards. According to the Nellie May CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Equations and Inequalities

Equations and Inequalities Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with

More information

Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook

Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

More information

Mth 95 Module 2 Spring 2014

Mth 95 Module 2 Spring 2014 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

Mathematics Placement

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

More information

MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

GRE MATH REVIEW #5. 1. Variable: A letter that represents an unknown number.

GRE MATH REVIEW #5. 1. Variable: A letter that represents an unknown number. GRE MATH REVIEW #5 Eponents and Radicals Many numbers can be epressed as the product of a number multiplied by itself a number of times. For eample, 16 can be epressed as. Another way to write this is

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

ModuMath Algebra Lessons

ModuMath Algebra Lessons ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

More information

Rational Expressions and Rational Equations

Rational Expressions and Rational Equations Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called. Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

More information

FACTORING ax 2 bx c WITH a 1

FACTORING ax 2 bx c WITH a 1 296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In

More information

Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

More information

Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

More information

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

More information

As evident above, you will need strong factoring skills to find the domain of many rational expressions.

As evident above, you will need strong factoring skills to find the domain of many rational expressions. You may write in this packet, but answer all questions on separate sheets of paper. A "rational epression" is a polynomial fraction. In general, what holds true for simple fractions is true for rational

More information

Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Exercises. For students who are entering. Pre-Calculus Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

More information

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.

Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations. Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable

More information

Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2 4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

SPECIAL PRODUCTS AND FACTORS

SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

Florida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper

Florida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,

More information

Quick Reference ebook

Quick Reference ebook This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Algebra 2 PreAP. Name Period

Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

Algebra 1 Course Objectives

Algebra 1 Course Objectives Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in

More information

Multiplying Polynomials 5

Multiplying Polynomials 5 Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive

More information

EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.

EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an

More information

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

More information

6.3 FACTORING ax 2 bx c WITH a 1

6.3 FACTORING ax 2 bx c WITH a 1 290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

More information

Substitute 4 for x in the function, Simplify.

Substitute 4 for x in the function, Simplify. Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The

More information

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

More information

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013 Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

More information

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

More information

A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated

A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means

More information

Simplifying Square-Root Radicals Containing Perfect Square Factors

Simplifying Square-Root Radicals Containing Perfect Square Factors DETAILED SOLUTIONS AND CONCEPTS - OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

More information

CHAPTER 7: FACTORING POLYNOMIALS

CHAPTER 7: FACTORING POLYNOMIALS CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Equations Involving Fractions

Equations Involving Fractions . Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation

More information

CAMI Education linked to CAPS: Mathematics

CAMI Education linked to CAPS: Mathematics - 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent. Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

More information

Algebra 1-2. A. Identify and translate variables and expressions.

Algebra 1-2. A. Identify and translate variables and expressions. St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

More information

Pre-Calculus II Factoring and Operations on Polynomials

Pre-Calculus II Factoring and Operations on Polynomials Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...

More information

Accuplacer Elementary Algebra Study Guide for Screen Readers

Accuplacer Elementary Algebra Study Guide for Screen Readers Accuplacer Elementary Algebra Study Guide for Screen Readers The following sample questions are similar to the format and content of questions on the Accuplacer Elementary Algebra test. Reviewing these

More information

MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

More information

TSI College Level Math Practice Test

TSI College Level Math Practice Test TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)

More information

Introduction to the Instructor TERM 1

Introduction to the Instructor TERM 1 Introduction to the Instructor TERM 1 This calendar of lessons was prepared as a textbook independent sequence of lessons and the order of topics can be modified based on the textbook selection. The columns

More information

Lesson 9: Radicals and Conjugates

Lesson 9: Radicals and Conjugates Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

More information

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

More information

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

More information

Skill Builders. (Extra Practice) Volume I

Skill Builders. (Extra Practice) Volume I Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?

SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3? SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information