Math 002 Intermediate Algebra Spring 2012 Objectives & Assignments

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1 Math 00 Intermediate Algebra Spring 01 Objectives & Assignments Unit 3 Exponents, Polynomial Operations, and Factoring I. Exponents & Scientific Notation 1. Use the properties of exponents to simplify expressions.. Write large/small numbers in scientific notation. 3. Expand scientific notation to write large/small numbers. II. Polynomial Operations 4. Use function notation with polynomials. 5. Add or subtract polynomials by combining like terms. 6. Perform multiplication of polynomials using the distributive property and special product patterns. 7. Factor polynomials completely. III. Applications of Factoring 8. Use the zero-factor property to solve polynomial equations. 9. Identify zeros from a graph and form polynomials equations by using the zero-factor property in reverse. Unit 3 Topic Homework Key Problems 5.1 Exponents & Scientific Notation pg. 331: 10, 11, 14, 16, 35, 39, 49-61(odd), 66, 69, 70, 81, 8, 94-10(even), 111, 114, 115, 118, , 61, 70, 8, 96, More pg. 338: 18-54(M3), 55, 57, 59, 63 4, 30, 48, 54, Polynomials & Polynomial Functions 5.4 Multiplying Polynomials 5.5 Factoring by GCF & Grouping pg. 351: 35-75(M5), 76, 84, 109, 110, 111, 113 pg. 361: 50, 53, 55, 60, 63, 65, 68, 70, 73, 80, 84, 85, 94, 96 45, 50, 75, 84, , 60, 63, 68, 85 pg. 366: 5-75(M5), 79, 89 5, 35, 40, 50, Factoring Trinomials pg. 376: 1-7(odd), 70, 71, 76, 78 5, 9, 19, 7, Factoring by Special Products 5.8 Solve by Factoring pg. 38: 6, 1, 15, 0, 3, 6, 35-38, 45, 49, 51, 54, 80, 81 pg. 396: 30-54(M3), 55, 75, 77, 81, 83, 89-94(all) 6, 45, 6, 54, 80 36, 48, 55, 77, 94 1

2 Math 00 Unit 3 - Student Notes Sections 5.1 & 5.- Exponents and Scientific Notation Objectives: Use the product rule for exponents. Use the quotient rule for exponents. Evaluate expressions raised to the zero power. Evaluate expressions raised to the negative nth power. Use the power rule for exponents. Use exponent rules and definitions to simplify exponential expressions. Convert between scientific notation and standard notation. 1. Define an exponent: What is the difference?. Section 5.1 Rules for Exponents: Using the product rule. EX 1, EX Product of Exponents: Generalize: *Note that the bases must be the same. a. b. c.

3 Using the quotient rule. Ex 4 Quotients of Exponents Generalize: *Note that the bases must be the same. d. e. f. g. Evaluating expressions raised to the 0 power. EX 3 Zero Exponent Consider the following problem: h. i. j. Evaluating exponents raised to the negative nth power. EX 5, EX 6 Negative Exponent Consider the following problem: k. l. m. n. 3

4 Simplify using the rules together. EX 5, EX 6 o. p. q. 3. Section 5. Rules for Exponents: Power of a Product/Quotient. EX, EX 3, EX 4, EX 5 Power Rule. EX 1 Simplifying exponents Putting it all together. r. s. t. u. v. w. **Properties of exponents can only be applied to expressions that have the same base. Consequently, the 3 expression x y cannot be further simplified. **Powers can only be distributed to factors, not terms. To simplify the expression ( x y ) 3 take the whole factor ( x y ) and square it. 3, you must 4

5 4. Converting between scientific notation and standard notation. Scientific notation: Scientific Notation provides a compact way of writing large and small numbers. A positive number is written in scientific notation if it is written as the product of a number a, where 1 a 10 and an integer power r of 10: Examples Non-Examples There are many ways of rewriting numbers. Scientific notation is just one particular way. Writing a number in scientific notation. EX 8 Step 1: Move the decimal point in the original number until the new number has a value between 1 and 10. Step : Step 3: Count the number of the decimal places the decimal point was moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative. Write the product of the new number in Step 1 and 10 raised to an exponent equal to the count in Step. Example 1: Write the numbers in scientific notation. a) 93,000,000 b) c) 17,500 d) Writing a scientific notation number in standard notation. EX 9 Move the decimal point in the number the same number of places as the exponent on 10. If the exponent is positive, the number is 10 or greater. If the exponent is negative, the number is less than 1. Example : Write the numbers in standard notation. a) b) c) d) 5

6 Section Polynomials and Polynomial Functions Objectives: Vocabulary is important: term, constant, polynomial, monomial, binomial, trinomial, degree, factor, product, variable, etc. Review combining like term. Add & subtract polynomials: vertical & horizontal. Define polynomial functions. We re not looking at graphs of polynomial functions in this section. 1. Vocabulary term: a number or the of a number and variables. Examples: Polynomial: A sum of where all variables have powers and no appear in the denominator. Circle the polynomial(s). If not a polynomial, explain why. A monomial is a polynomial consisting of. EX A binomial is a polynomial consisting of. EX A trinomial is a polynomial consisting of. EX The degree of a term is the of the exponents of its variables. EX 1 Ex 1: 5. x, 3x, 53, 7g p, 14st The degree of a polynomial is determined by the with the degree. EX, EX 3 Ex : 14st 1s t, 15p 5p 3p 4, 17a b 1a b A polynomial is written in descending order when it is written with. Ex 3: Given 5 4x 3x write in descending order. 6

7 . Review like terms. Recall that like terms have the same variables with the same exponents. Like Terms Unlike Terms 3. Adding and Subtracting monomials: simply combine like terms. EX 6 Ex 4 a. 4. Adding and Subtracting Polynomials. EX 7, EX 8, EX 9, EX 10 Horizontal method Vertical method Ex 5 a) ( a a 4) (5 a 6a 5) b) ( 4q 7) ( q 8q 9) c) d) Subtract from Make sure to discuss order in subtraction. 5. Defining polynomial functions: We will use P(x) to represent a polynomial function. EX 4 Ex 6 Given P( x) 3x x, find the following. a) P( ) b) P( 1 ) 7

8 Section 5.4 Multiplying Polynomials Objectives: Multiplying polynomials: o Distributive property o FOIL Method o Box Method o Special Products Evaluate polynomial functions. 1. Multiplying Two Polynomials. Using the distributive property. EX 1, EX, EX 3 3 a) p ( 3p 5p) b) ( a 4)( 3a 5) Using the FOIL method. EX 5, EX 6 a) ( a 1)( 3a ) b) (5 y z)( y 3 z) Using the Box method. c) ( 3x )( x 4x 5) d) ( x 3)( x 4x 1) 8

9 . Special Products. Square of a binomial. EX 7 ( x 3y) = ( x y) ( 3a b) ( x y) ( x y) ( 3t ) Sum and difference of two terms: conjugates. EX 8 ( x y)( x y) ( b )( b ) ( 6p 5q)( 6p 5q) 3. Evaluating polynomial functions. EX 4 Given, find the following: = 9

10 Section 5.5 The Greatest Common Factor and Factoring by Grouping Objectives: Identify the GCF. Factor out the GCF of a polynomial s terms. Factor by grouping. Warm-up: Multiply: Factoring is the reverse process of multiplying. It is the process of writing a polynomial as a product. 1. Finding the GCF, Greatest Common Factor, from a list of monomials. EX 1 Step 1: Find the GCF of the numerical coefficients. Step : Step 3: Ex 1: Find the GCF of the variable factors. The product of the factors found in Steps1 and is the GCF of the monomials. Find the GCF a) 3x y 3, 9x 4 y 3 b) 6a, 1a 3 b, 36ab. Factoring out the GCF of a polynomial s terms. Use the distributive property in reverse. EX EX 5 a) common monomial factor: factor x out of each term Use the distributive property in reverse. b) common monomial factor: factor 4x out of each term Use the distributive property in reverse. 3. Factor out -1 from a polynomial. a) x y b) 3x 8z c) 3r s 3 10

11 4. Factoring Polynomials by Grouping. Final answer must be a product. EX 7 EX 10 a) 15 0x 6y 8 xy b) 0xy 8x 5y ab d) xy 4y 3x 1 c) 16 8a 6b 3 FACTORING BY GROUPING Sometimes it is possible to factor a polynomial by grouping terms of the polynomial and looking for common terms in each group. Factor out the GCF of the polynomial. Group the polynomial into two binomials. Factor out the GCF from each binomial. Factor out the GCF of the entire polynomial. 11

12 Section 5.6 Factoring Trinomials Objectives: Factor trinomials of the form. Factor trinomials of the form using the ac method. Review. Recall that to multiply x and x 3, we proceed as follows: 1. Factoring trinomials of the form : Leading coefficient 1. EX 1, EX, EX 3 Find all possible factor pairs of the constant. Find the pair of factors whose sum is. Example 1 Factor each trinomial if possible. a) m 3m 10 factors of : sum of factors = b) factors of : sum of factors = c) 3y 1y 18 d) FACTORING A TRINOMIAL OF THE FORM x bx c If the trinomial has a leading coefficient of 1, follow these steps to factor: find all possible factor pairs of the constant c find the pair of factors whose sum is the middle term b 1

13 . Factoring trinomials of the form : Leading coefficient is not 1. EX 9, EX 10 Use the method: Step 1: Find two numbers whose product is a c and whose sum is b. Step : Write the term bx as a sum by using the factors found in Step 1. Step 3: Factor by grouping. Example Factor each trinomial if possible. a) b) c) d) 13

14 Section 5.7 Factoring by Special Products Objectives: Factor a perfect square trinomial. Factor the difference of two squares. Factor the sum or difference of two cubes. Some trinomials have special forms. If you can recognize them, factoring will be easier. 1. Square Trinomials: When you square binomials, the resulting polynomials are called perfect square trinomials.. Factoring a perfect square trinomial. EX 1, EX Example 1: Factor completely. a) x 30x 5 b) x x 11 c) Generalize: 14

15 3. Factoring the difference of two squares. EX 3, EX 4 Recall: The product of conjugate pairs is the difference of squares. Example : Factor completely. d) e) f) g) 4. Factoring the sum of two squares. 5. Factoring the sum and difference of two cubes. EX 6, EX 7, EX 8, EX 9 S O A P x 3 y 3 x 3 y 3 Example 3: h) i) j) 15

16 Steps for factoring a polynomial 1. Factor out all common factors.. If a polynomial has two terms, check for the following types: a. The difference of two squares: b. The sum of two squares: c. The sum of two cubes: d. The difference of two cubes: 3. If a polynomial has three terms, check for the following problem types: a. A perfect square trinomial: b. If the trinomial is not a perfect square, attempt to factor it as a general trinomial using factoring by grouping. 4. If a polynomial has four or more terms, try factoring by grouping. 5. Continue until each individual factor is prime. 6. Check the results by multiplying. Factoring Flowchart: 16

17 Section 5.8 Solving Equations by Factoring and Problem Solving Objectives: Solve polynomial equations by factoring. Find the x-intercepts of a polynomial function. Solve problems that can be modeled by polynomial equations. 1. Vocabulary: A polynomial equation is the result of setting two polynomials equal. A polynomial equation is in standard form if one side of the equation is.. Zero-Factor Property. EX 1 EX 7 Zero-Factor Property If A and B are real numbers and, then or This property is true for three or more factors also. In other words, if the product of two or more real numbers is zero, then at least one number must be zero. This is the main idea of this section and a key concept in algebra. Example 1: Solve. 17

18 3. Solving a Polynomial Equation by factoring: EX 1 EX 7 Example : Solve. a. b. Generalize Step 1: Step : Write the equation in standard form. Factor the polynomial completely. Step 3: Set each factor containing a variable equal to 0. Step 4: Step 5: Solve the resulting equations. Check each solution in the original equation. 4. Finding -intercepts of polynomial functions. (a) Match each function with its graph. EX 10 Solutions to a polynomials equation are also called. Graphically, solutions to a polynomial function are where the graph or rather,. (b) Write the factored form of an equation given solutions. Example 3: Write the factored form of an equation which has the solutions -4, -1, and 8. 18

19 5. Solving problems modeled by polynomial equations. (a) Using the Pythagorean Theorem. EX 9 (#78) The longer leg of a right triangle is 4 feet longer that the other leg. Find the length of the two legs if the hypotenuse is 0 feet. (b) Finding the return time of a rocket. EX 8 After t seconds, the height of a model rocket launched from the ground into the air is given by the function. 1) How high is the rocket at second? ) Find how long it takes the rocket to reach a height of 96 feet. 3) When will the rocket hit the ground? 19