Algebra 1CP Final. Name: Class: Date:

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1 Name: lass: ate: I: lgebra 1P Final Multiple hoice Identify the choice that best completes the statement or answers the question. Ï 3x 3y Tell whether the ordered pair ( 4, 5) is a solution of the system Ô Ì. ÓÔ 3x + y 8 a. no b. yes Ï x y 4 2. Solve Ô Ì by substitution. Express your answer as an ordered pair. ÓÔ x 2y 12 a. ( 6, 0) c. (8, 0) b. (0, 8) d. (0, 8) Ï 9x + 2y Solve Ô Ì by elimination. Express your answer as an ordered pair. ÓÔ 7x + 2y 12 a. (5, -5) c. ( 12, 48) b. (0, 5) d. (5, -5) 4. t the local pet store, zebra fish cost $2.20 each and neon tetras cost $1.75 each. If Sameer bought 17 fish for a total cost of $33.80, not including tax, how many of each type of fish did he buy? a. 9 zebra fish, 8 neon tetras c. 6 zebra fish, 11 neon tetras b. 8 zebra fish, 9 neon tetras d. 11 zebra fish, 6 neon tetras 5. n airplane travels 900 miles from Houston to Miami in 6 hours against the wind. On its return trip, with the wind, it takes only 5 hours. Find the rate of the airplane with no wind. Find the rate of the wind. a. The airplane flies at 155 mi/h with no wind. The rate of the wind is 135 mi/h. b. The airplane flies at 165 mi/h with no wind. The rate of the wind is 135 mi/h. c. The airplane flies at 155 mi/h with no wind. The rate of the wind is 15 mi/h. d. The airplane flies at 165 mi/h with no wind. The rate of the wind is 15 mi/h. 6. paint mixer wants to mix paint that is 15% gloss with paint that is 30% gloss to make 6 gallons of paint that is 25% gloss. How many gallons of each paint should the paint mixer mix together? a. The paint mixer should use 4 gallons of 15% gloss paint and 2 gallons of 30% gloss paint.. b. The paint mixer should use 3 gallons of 15% gloss paint and 3 gallons of 30% gloss paint. c. The paint mixer should use 2 gallons of 15% gloss paint and 4 gallons of 30% gloss paint. d. The paint mixer should use 1 gallon of 15% gloss paint and 5 gallons of 30% gloss paint. 7. Tell whether (3, 2) is a solution of y > 4x 5. a. No, (3, 2) is not a solution of y > 4x 5. b. Yes, (3, 2) is a solution of y > 4x 5. 1

2 Name: I: 8. Write an inequality to represent the graph. a. y 3x + 2 c. y > 3x + 2 b. y 2x + 3 d. y 3x Simplify 4 2. a. 8 c b. 16 d. 10. Simplify ( 4) a. 1 c. 1 4 b. 4 d Find the value of the power a. 10 c. 100 b d Find the value of the expression a c. 5,220 b d Simplify a. 12 c b. d. annot simplify Simplify m 9 y 4 m 9. a. m 0 y 4 c. m 81 y 4 b. (m y) 22 d. m 18 y Simplify (x 5 ) 8 x 3. 1 a. c. x 37 b. x 37 d. 1 x 9 1 x 120 2

3 Name: I: 16. Simplify a. annot simplify c. 1,296 b. 46,656 d. 5 Ê 4r 8 2 ˆ 17. Simplify Ë Á r 5 s 3 a. b.. 16 r 5 s 6 c. 8r 10 s 5 d. 18. Simplify the expression 32. a. 16 c. 20 b. 13 d r Tell whether the number 7 is a root of 2r 2 11r 63. a. Yes b. No 20. dd or subtract. 3z 2 z 4 + 5z z 4 a. 21z 6 c. 8z z 8 b. 2z 2 15z 4 d. 8z z Subtract. (8b 4 b 3 ) (b 4 + 3b 3 1) a. 8b 4 4b 3 1 b. 8b 4 + 3b 3 1 c. 7b 4 4b 3 d. 7b 4 4b Multiply. 7x y 2Ê x 3 y 4 4x y 3 ˆ Ë Á a. 7x 4 y x 2 y 5 c. 7x 3 y x 1 y 6 b. 6x 4 y 6 11x 2 y 5 d. 7x 5 y 7 7x 3 y Multiply. (n 4)(n + 1) a. n 2 4 c. n 2 4n 4 b. n(n + 1) 4(n + 1) d. n 2 3n Multiply. (m 7) 2 a. p c. m m + 49 b. p 2 64 d. m 2 14m + 49 s 6 8r 6 s 6 3

4 Name: I: 25. Multiply. (r + 6)(r 6) a. r 2 6r + 36 c. 2r 12 b. r 2 36 d. r Factor the polynomial 15x x 3 6x 2. a. 3(5x x 3 2x 2 ) b. annot be factored c. x 2 (15x x 6) d. 3x 2 (5x x 2) 27. Factor the trinomial d 2 + 2d 48. a. (d 6)(d + 8) c. (d 1)(d 48) b. (d 8)(d 6) d. (d + 1)(d 48) 28. Factor 3x x 25. a. (x 5)(3x + 5) c. (x 5)(3x 5) b. (x + 5)(3x + 5) d. (x + 5)(3x 5) 29. Without graphing, tell whether the point ( 3, 4) is on the graph of y 3x 2 8. a. No b. Yes 4

5 Name: I: 30. Graph y x 2 3x + 4. Find the axis of symmetry and the vertex. a. c. b. The axis of symmetry is x 0. The vertex is (0, 4). d. The axis of symmetry is x 4. The vertex is (4, 0). The axis of symmetry is x 3. The 2 The axis of symmetry is x 3. The 2 Ê 3 vertex is ˆ Ê 2 4 Ë Á. vertex is 3, 7 ˆ 2 4 Ë Á. 31. Solve the quadratic equation b 2 8b by factoring. a. 3 and 5 c. 3 and 5 b. 3 and 5 d. 3 and Solve x 2 64 by using square roots. a. The solutions are 8 and 8. c. The solution is 8. b. The solutions are 4096 and d. The solution is Solve 100x by using square roots. a. ± 100 c. ± b. ± d. No solution 34. omplete the square for x 2 14x +? to form a perfect square trinomial. a. x 2 14x 49 c. x 2 14x + 49 b. x 2 14x d. x 2 14x 196 5

6 Name: I: 35. Solve 2x 2 x 4 0 by using the Quadratic Formula. a. x 1 ± 33 4 b. no solution d. x c. x 1 ± ± Simplify the rational expression 5t3 10t. Identify any excluded values. t 2 2t a. 5t; t 2 or 0 c. 5t; t 2 Ê b. 5t t 2 ˆ 2t Ë Á ; t 2 or 0 d. 5t; no excluded values 37. Multiply. Simplify your answer. x 2 x 6 2x 2 6x x 2 + x x 2 + 4x + 4. a. x + 1 2x + 4 b. x x ivide. Simplify your answer. 1 n n 7 7n 1(7n) a. n( n 7) n 7 b dd. Simplify your answer. 3x + 8 x x a. x + 4 3( x + 4) b. x dd. Simplify your answer. 3y 9y 3y 2 18y a. b. 1 3 y + 2 6y c. d. c. d. c. d. c. d. 2x 2 6 3x 2 2x n 7 7 n 3 x 4 3 2( x 4) y 2 6y 1 y 6

7 Name: I: 41. ivide. Simplify your answer. (12x 4 18x x 2 ) 6x 3 a. 12x x c. 2x 4 3x 3 + 6x 2 b. 2x d. 6x 12 x x x ivide. (2x 2 + 3x 15) (x 3) a. 2x 3 6 c. 2x x 3 x Solve b. 2x + 21 d. x x m 3 3. heck your answer. 5m a. m 9 17 c. m 9 23 b. m 9 17 d. m Solve 6 t heck your answer. 4 a. t2 c. t-12 b. t12 d. t Simplify the expression 16x 5 y 2. ll variables represent nonnegative numbers. a. 4x 2 y x c. 4x 4 y 2 x b. 4x 2 y x 2 d. 4 x Simplify b 9. The variable represents a nonnegative number. 16b a. b. b 8 4 b Simplify a. b c. d. c. d. b 4 4 b Simplify the expression 75b b 2 27b. Ê ˆ a. 147b c Ë Á b b. 7 3b d. 12 3b

8 Name: I: 49. Multiply. Write the product in simplest form. 25h 10h a. 4b 15 c. 30b b. 2b 15 d. b Simplify the quotient a. b Solve the equation b 18. heck your answer. a. b 324 c. b 36 b. b 324 d. b Solve 2y heck your answer. c. d a. y 9 2 c. y 2 9 b. No solution. d. y

9 I: lgebra 1P Final nswer Section MULTIPLE HOIE 1. NS: Substitute 4 for x and 5 for y in both equations. Since these values make the second equation false, ( 4, 5) is not a solution of the system. Use substitution to check that the ordered pair satisfies both equations. PTS: 1 IF: asic REF: Page 329 OJ: Identifying Solutions of Systems ST: 19.0 TOP: 6-1 Solving Systems by Graphing KEY: ordered pair system of equations solution NT: g 1

10 I: 2. NS: Step 1 x 6y + 48 Solve the second equation for x. Step 2 3( 6y + 48) 2y 16 Substitute 6y + 48 for x in the first equation. Step 3 18y y 16 Use the istributive Property to simplify. 20y ollect like terms. 20y 16 (144) 20y 160 Subtract 144 from both sides. ivide both sides by 20. y 8 Step 4 x + 6y 48 x + 6(8) 48 x x 48 (48) x 0 Write one of the original equations. Substitute 8 for y. Subtract 48 from both sides. Step 5 (0, 8) Write the solution as an ordered pair. fter solving one equation for a variable, substitute the value into the other original equation, not the one that has just been solved. heck the signs. fter solving one equation for a variable, substitute the value into the other original equation, not the one that has just been solved. PTS: 1 IF: verage REF: Page 338 OJ: Using the istributive Property NT: e ST: 19.0 TOP: 6-2 Solving Systems by Substitution 2

11 I: 3. NS: 9x + 2y 12 7x 2y 12 Multiply all expressions in the second equation by 1. 2x + 0y 24 dd the two equations together. x 24 2 ivide both sides by 2. x 12 Solve for x ( 12) y 2 y 48 Substitute the value for x into one of the original equations and solve for y. Multiply all terms in the second equation by -1 before combining the equations. Multiply all terms in the second equation by -1 before combining the equations. Multiply all terms in the second equation by -1 before combining the equations. PTS: 1 IF: verage REF: Page 344 OJ: Elimination Using Subtraction ST: 19.0 TOP: 6-3 Solving Systems by Elimination KEY: system of equations elimination NT: g 3

12 I: 4. NS: Let z be the number of zebra fish and let n be the number of neon tetras that Sameer bought. Then solve the following system of equations. 2.20z n Sameer spent $ z + n z n Sameer bought 17 fish. 2.20z 2.20n Multiply the second equation by n 3.60 dd the two equations to eliminate the z term. n 8 Solve for n. To solve for z, substitute 8 for n in the first equation. 2.20z ( 8) z 19.8 z 9 Simplify. Solve for z. You switched the prices of zebra fish and neon tetras. Write an equation expressing the total cost and a second equation expressing the total number of fish. Solve for z and n using elimination. Write an equation expressing the total cost and a second equation expressing the total number of fish. Solve for z and n using elimination. PTS: 1 IF: verage REF: Page 346 OJ: pplication NT: g ST: 7F1.1 TOP: 6-3 Solving Systems by Elimination 4

13 I: 5. NS: Let p be the rate at which the airplane travels with no wind, and let w be the rate of the wind. Use a table to set up two equations one for the trip against the wind and one for the trip with the wind. Rate Time istance With the wind p w gainst the wind p + w Ï Ï 6(p w) 900 Solve the system Ô Ì. First write the system as Ô 6p 6w 900 Ì, and then use elimination. ÓÔ 5(p + w) 900 ÓÔ 5p + 5w 900 Step 1 5(6p 6w 900) Multiply each term in the first equation by 6(5p + 5w 900) 5 and each term in the second equation by 6. 30p 30w p + 30w 5400 dd the new equations. Step 2 60p 9900 p 165 Simplify and solve for p. Step 3 6p 6w 900 Write one of the original equations. 6(165) 6w 900 Substitute 165 for p w Subtract 990 from both sides. 6w 90 Simplify and solve for w. w 15 Step 4 (165,15) Write the solution as an ordered pair. The airplane flies at 165 mi/h with no wind. The rate of the wind is 15 mi/h. Use a table to set up two equations one for the trip against the wind and one for the trip with the wind. heck your math. Use a table to set up two equations one for the trip against the wind and one for the trip with the wind. PTS: 1 IF: dvanced REF: Page 356 OJ: Solving Rate Problems ST: TOP: 6-5 pplying Systems 5

14 I: 6. NS: Let x be the number of gallons of 15% gloss paint, and let y be the number of gallons of 30% gloss paint. Use a table to set up two equations one for the amount of gloss in the first paint, and one for the amount of gloss in the second paint. 15% gloss + 30% gloss 25% gloss mount of paint (gallons) x + y 6 mount of gloss (gallons) 0.15x y 0.25(6) 1.5 Ï x + y 6 Solve the system Ô Ì. Use substitution. ÓÔ 0.15x y 1.5 Step 1 x + y 6 Solve the first equation for x by subtracting y y y from both sides. x 6 y Step x y 1.5 Substitute 6 y for x in the second equation. 0.15(6 y) y (6) 0.15(y) y 1.5 istribute 0.15 to the expression in y y 1.5 parentheses. Step y 1.5 Simplify. Solve for y Subtract 0.90 from both sides. 0.15y y ivide both sides by y 4 Step 4 x + y 6 Write one of the original equations. x Substitute 4 for y. 4 4 Subtract 4 from both sides. x 2 Step 5 (2, 4) Write the solution as an ordered pair. The paint mixer should use 2 gallons of 15% gloss paint and 4 gallons of 30% gloss paint. Use a table to set up two equations one for the amount of gloss in the first paint, and one for the amount of gloss in the second paint. Use a table to set up two equations one for the amount of gloss in the first paint, and one for the amount of gloss in the second paint. Use substitution to solve the system of equations. PTS: 1 IF: dvanced REF: Page 357 OJ: Solving Mixture Problems ST: TOP: 6-5 pplying Systems 6

15 I: 7. NS: Substitute (3, 2) for (x, y) in y > 4x 5. y > 4x 5 2 > 4(3) 5 2 > 7, false (3, 2) is not a solution of y > 4x 5. Substitute the values for (x, y) into the inequality to see if the ordered pair is a solution. PTS: 1 IF: asic REF: Page 364 OJ: Identifying Solutions of Inequalities NT: a ST: 16.0 TOP: 6-6 Solving Linear Inequalities 8. NS: Use the graph to determine the slope and y-intercept, and then write an equation in the form y mx + b. graph shaded above the line means greater than and the graph shaded below the line means less than. Use or if the line is solid; use < or > if the line is dashed. Use the graph to find the slope and y-intercept. Then write an equation for the boundary line in the form y mx + b, where m is the slope and b is the y-intercept. Use "greater than or equal to" or "less than or equal to" for a solid line. Use "greater than" or "less than" for a dashed line. heck the direction of the inequality symbol. PTS: 1 IF: verage REF: Page 367 OJ: Writing an Inequality from a Graph ST: 19.0 TOP: 6-6 Solving Linear Inequalities KEY: graph inequality equation of a line NT: d 7

16 I: 9. NS: The reciprocal of 4 is nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the opposite (positive) exponent. nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the opposite (positive) exponent. heck the sign of your answer. negative exponent does not affect the sign of the answer. PTS: 1 IF: verage REF: Page 395 OJ: Zero and Negative Exponents NT: d ST: 7F2.1 TOP: 7-1 Integer Exponents KEY: negative exponent evaluate power exponent 10. NS: ny nonzero base to the zero power is equal to 1. ( 4) 0 1 nonzero number raised to the zero power is equal to 1. nonzero number raised to the zero power is equal to 1. nonzero number raised to the zero power is equal to 1. PTS: 1 IF: verage REF: Page 395 OJ: Zero and Negative Exponents NT: d ST: 7F2.1 TOP: 7-1 Integer Exponents KEY: zero exponent zero power evaluate power exponent 11. NS: Start with 1 and move the decimal point two places to the right ount the number of places to move the decimal point. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left. Start with 1 and move the decimal point. PTS: 1 IF: asic REF: Page 400 OJ: Evaluating Powers of 10 NT: f ST: 7F2.1 TOP: 7-2 Powers of 10 and Scientific Notation 8

17 I: 12. NS: Move the decimal point 3 places to the left Move the decimal point the correct number of places. Move the decimal point the correct number of places. For powers of 10, the exponent tells the number of places to move the decimal point. PTS: 1 IF: verage REF: Page 401 OJ: Multiplying by Powers of 10 NT: f ST: 7NS1.1 TOP: 7-2 Powers of 10 and Scientific Notation KEY: exponents multiplication power powers of NS: To multiply powers with the same base, keep the same base and add the exponents. Then, evaluate the power The exponent tells how many times to multiply the base number by itself. heck the sign of the exponent. If the bases are the same, add the exponents. Then evaluate the power. PTS: 1 IF: asic REF: Page 409 OJ: Finding Products of Powers NT: c ST: 12.0 TOP: 7-3 Multiplication Properties of Exponents KEY: evaluate product multiply power exponent 14. NS: To multiply powers with the same base, keep the same base and add the exponents. m 9 y 4 m 9 (m 9 m 9 ) y 4 m 18 y 4 To multiply powers with the same base, add the exponents, not subtract. Rewrite only powers with the same base. o not combine powers with different bases. To multiply powers with the same base, add the exponents, not multiply. PTS: 1 IF: verage REF: Page 409 OJ: Finding Products of Powers NT: c ST: 12.0 TOP: 7-3 Multiplication Properties of Exponents KEY: evaluate product multiply power exponent 9

18 I: 15. NS: (x 5 ) 8 x 3 x 5( 8) x 3 Use the Power of a Power Property. x 40 x 3 x x 37 1 x 37 Simplify the exponent of the first term. dd the exponents since the powers have the same base. Write with a positive exponent. ontinue simplifying to a fraction with a positive exponent. Multiply the exponents when a power is raised to another power. dd the exponents when a power is multiplied by another power. Multiply the exponents when a power is raised to another power. PTS: 1 IF: dvanced REF: Page 410 OJ: Finding Powers of Powers NT: c ST: 12.0 TOP: 7-3 Multiplication Properties of Exponents 16. NS: To divide powers with the same base, keep the same base and subtract the exponents. The bases are the same, so the expression can be simplified. To divide powers with the same base, subtract the exponents. To divide powers with the same base, subtract the exponents. PTS: 1 IF: asic REF: Page 415 OJ: Finding Quotients of Powers ST: 12.0 TOP: 7-4 ivision Properties of Exponents KEY: exponent power division base 10

19 I: 17. NS: 4r 8 2 Ê ˆ Ë Á r 5 s 3 Ê Ë Á 4r 3 s 3 Ê Ë 4r 3 Á Ê Ë s 3 Á ˆ 2 ˆ 2 Simplify exponents with like bases: r8 r 5 r3. Use the Power of a Quotient Property. 2 ˆ ( 4) 2 (r 3 ) 2 Use the Power of a Product Property. (s 3 ) 2 (16)(r 3 ) 2 (s 3 ) 2 Simplify: ( 4) r 6 s 6 Use the Power of a Power Property to simplify the exponents. heck to see if the terms are correctly placed in the numerator and denominator. Use the Power of a Power Property to raise every term in the problem to the exponent outside the parenthesis. Use the Power of a Power Property to raise the constant to the power outside the parenthesis. PTS: 1 IF: verage REF: Page 417 OJ: Finding Positive Powers of Quotients NT: c ST: 12.0 TOP: 7-4 ivision Properties of Exponents 11

20 I: 18. NS: Ê 5 ˆ 32 Ë Á 4 Ê ˆ Ë Á ( 2) efinition of b m n. number raised to the power of m/n is equal to the nth root of the number raised to the mth power. Rewrite the base as a number raised to a power. Rewrite the denominator of the power as a root. PTS: 1 IF: verage REF: Page 423 OJ: Simplifying Expressions with Fractional Exponents ST: 12.0 TOP: 7-5 Fractional Exponents KEY: fractional exponent 19. NS: 2r 2 11r 63 2(7) 2 11(7) 63 Substitute for r Simplify. 42 No! 42 0, so 7 is not a root of 2r 2 11r 63. root of a polynomial in one variable is a value of the variable for which the polynomial is equal to 0. PTS: 1 IF: verage REF: Page 432 OJ: Identifying Roots of Polynomials ST: TOP: 7-6 Polynomials KEY: polynomial 12

21 I: 20. NS: 3z 2 z 4 + 5z z 4 3z z 4 + 5z z 4 Identify like terms. Use the ommutative Property to 3z 2 + 5z 2 z z 4 move like terms together. 8z z 4 ombine like terms. Only add or subtract coefficients on like terms. heck your addition and subtraction. When combining like terms, only add or subtract the coefficients. The powers stay the same. PTS: 1 IF: dvanced REF: Page 438 OJ: dding and Subtracting Monomials NT: c ST: TOP: 7-7 dding and Subtracting Polynomials KEY: monomial 21. NS: (8b 4 b 3 ) (b 4 + 3b 3 1) (8b 4 b 3 ) + ( b 4 3b 3 + 1) Rewrite subtraction as addition of the opposite. (8b 4 b 4 ) + ( b 3 3b 3 ) + ( 1) Identify like terms. Rearrange terms to get like terms together. 7b 4 4b ombine like terms. heck the coefficients and the signs. First, rewrite the subtraction as an addition of the opposite. Then, combine the like terms. heck that you have included all the terms. PTS: 1 IF: verage REF: Page 439 OJ: Subtracting Polynomials NT: c ST: TOP: 7-7 dding and Subtracting Polynomials KEY: polynomial 13

22 I: 22. NS: Use the istributive Property to multiply the monomial by each term inside the parentheses. Group terms to get like bases together, and then multiply. Multiply the coefficients for each term; don't add. When multiplying like bases, add the exponents. on't forget to multiply the coefficients for each term. PTS: 1 IF: dvanced REF: Page 446 OJ: Multiplying a Polynomial by a Monomial NT: c ST: TOP: 7-8 Multiplying Polynomials KEY: polynomial multiplication 23. NS: (n 4)(n + 1) Use FOIL. n(n + 1) 4(n + 1) istribute n and 4. n(n) + n(+1) 4(n) 4(+1) istribute n and 4 again. n 2 + n 4n 4 Multiply. n 2 3n 4 ombine like-terms. You did not multiply the inner and outer terms. istribute again, multiply and combine like-terms. You did not multiply the outer terms. PTS: 1 IF: asic REF: Page 448 OJ: Multiplying inomials NT: c ST: TOP: 7-8 Multiplying Polynomials KEY: binomial multiplication 24. NS: (m 7) 2 ( a b) a 2 2ab + b 2 Use the rule for ( a b). (m 7) 2 m 2 2(m)(7) Use the FOIL method, and then combine like terms. m 2 14m + 49 Simplify. The last term in the product should be the square of the second term in the binomial. heck the signs. heck the signs. PTS: 1 IF: asic REF: Page 456 OJ: Finding Products in the Form (a b)^2 NT: c ST: TOP: 7-9 Special Products of inomials KEY: binomial multiplication 14

23 I: 25. NS: (r + 6)(r 6) ( a + b) ( a b) a 2 b 2 Use the rule for ( a + b) ( a b). (r + 6)(r 6) r Use the FOIL method, and then combine like terms. r 2 36 Simplify. First, use the FOIL method. Then, combine the like terms. Use the FOIL method. The terms in the product should be squares. PTS: 1 IF: asic REF: Page 457 OJ: Finding Products in the Form (a + b)(a b) NT: c ST: TOP: 7-9 Special Products of inomials KEY: binomial multiplication 26. NS: 15x x 3 6x 2 Find the GF. The GF of 15x 4, 36x 3, and 6x 2 is 3x 2. 3x 2 (5x 2 ) + 3x 2 (12x) 3x 2 (2) Write the terms as products using the GF. 3x 2 (5x x 2) Use the istributive Property to factor out the GF. First, find the GF. Then, write the terms as products using the GF. First, find the GF. Then, use the istributive Property to factor out the GF. First, find the GF. Then, write the terms as products using the GF. PTS: 1 IF: verage REF: Page 487 OJ: Factoring by Using the GF NT: b ST: TOP: 8-2 Factoring by GF 27. NS: d 2 + 2d 48 (d +? )(d +? ) Look for the factors of 48 whose sum is 2. (d 6)(d + 8) The factors are 6 and 8. Use the FOIL method to check your answer. heck the signs. Use the FOIL method to check your answer. PTS: 1 IF: asic REF: Page 498 OJ: Factoring x^2 + bx + c When c Is Negative ST: TOP: 8-3 Factoring x^2 + bx + c NT: d 15

24 I: 28. NS: Try factors of 3 for the coefficients and factors of 25 for the constant terms. The combination that works is: (x + 5)(3x 5) 3x 2 5x + 15x 25 3x x 25 Multiply the factors to check your answer. heck the signs. Multiply the factors to check your answer. PTS: 1 IF: asic REF: Page 505 OJ: Factoring ax^2 + bx + c NT: d ST: TOP: 8-4 Factoring ax^2 + bx + c KEY: factor trinomial guess and check 29. NS: Substitute ( 3, 4) into y 3x 2 8. y 3x 2 8 4? 3( 3) 2 8 4? ? No! No, since ( 3, 4) is not a solution of y 3x 2 8, ( 3, 4) is not on the graph. Substitute the x-coordinate of the point into the equation and check if the solution is equal to the corresponding y-coordinate. PTS: 1 IF: verage REF: Page 544 OJ: etermining Whether a Point Is on a Graph ST: TOP: 9-1 Quadratic Equations and Functions 16

25 I: 30. NS: Step 1: Find the axis of symmetry. x ( 3) Use x b. Substitute 1 for a and 3 for b. 2a Step 2: Find the vertex. y x 2 3x + 4 Ê y 3 2 Ë Á ˆ 2 Ê Ë Á ˆ + 4 Substitute 3 for x. 2 y 7 The y-coordinate is Ê 3 The vertex is, 7 ˆ 2 4 Ë Á. Graph the equation correctly. heck the vertex. To find the axis of symmetry, substitute for a and b in b/2a. PTS: 1 IF: dvanced ST: TOP: 9-3 Graphing Quadratic Functions KEY: graph quadratic 31. NS: b 2 8b ( b 3) ( b 5) 0 Factor the trinomial. b 3 0 or b 5 0 Use the Zero Product Property. b 3 or b 5 Solve each equation. The solutions are 3 and 5. heck your factorization by multiplying the factors together. heck your factorization by multiplying the factors together. Substitute the solutions into the original equation to check your answer. PTS: 1 IF: verage REF: Page 577 OJ: Solving Quadratic Equations by Factoring NT: a ST: TOP: 9-5 Solving Quadratic Equations by Factoring 17

26 I: 32. NS: Take the square root of both sides of the equation. Remember that there are both positive and negative solutions to the square root. heck your answers by substituting into the original equation. Every positive number has two square roots, and negative numbers have no square roots that are real numbers. Every positive number has two square roots, and negative numbers have no square roots that are real numbers. PTS: 1 IF: asic REF: Page 582 OJ: Using Square Roots to Solve x^2 a NT: a ST: 12.0 TOP: 9-6 Solving Quadratic Equations by Using Square Roots 33. NS: 100x dd 121 to both sides. 100x ivide both sides by 100. x Take the square root of both sides. x 11 ± 10 Use the plus/minus sign to show positive and negative roots. The solutions are and If possible, take the square root to find x. heck your solution. heck your positive and negative signs. PTS: 1 IF: verage REF: Page 583 OJ: Using Square Roots to Solve Quadratic Equations NT: a ST: 12.0 TOP: 9-6 Solving Quadratic Equations by Using Square Roots 18

27 I: 34. NS: ivide the coefficient of the x-term by 2, and then square it to get the constant term, dd the result to the expression to form a perfect square trinomial. Ê Ë Á ˆ. The final term of a perfect square trinomial must be added, not subtracted. ivide the coefficient of the x-term by 2, and then square it to get the constant term. ivide the coefficient of the x-term by 2, and then square it to get the constant term. PTS: 1 IF: asic REF: Page 591 OJ: ompleting the Square NT: b ST: TOP: 9-7 ompleting the Square 35. NS: Write the equation in standard form: x 2 2x 8 0. Substitute for a, 2 for b, and 8 for c in the Quadratic Formula, x b ± b 2 4ac. Write as two equations, one adding the square root and one subtracting it. 2a Simplify each equation to find the values of x. For the numerator in the Quadratic Formula, find the square root before adding it to or subtracting it from b. The denominator of the Quadratic Formula is 2 times a. Use the opposite of b, the coefficient of the x-term in the equation. PTS: 1 IF: verage REF: Page 599 OJ: Using the Quadratic Formula NT: a ST: TOP: 9-8 The Quadratic Formula 36. NS: Factor common factors out of the numerator and/or denominator. ivide out the common factors to simplify the expression. Finally, use the original denominator to determine excluded factors. ivide out common factors. etermine excluded values from the original denominator. etermine excluded values from the original denominator. PTS: 1 IF: asic REF: Page 643 OJ: Simplifying Rational Expressions NT: c ST: TOP: 10-3 Simplifying Rational Expressions 19

28 I: 37. NS: x 2 x 6 2x 2 6x x 2 + x x 2 + 4x + 4 (x + 2)(x 3) x(x + 1) 2x(x 3) (x + 2)(x + 2) 1 (x + 1) 2 (x + 2) x + 1 2x + 4 Factor the numerator and denominator. Simplify. Multiply the remaining factors. Factor the numerator and denominator and divide out the common factors. Factor the numerator and denominator and divide out the common factors. Factor the numerator and denominator and divide out the common factors. PTS: 1 IF: verage REF: Page 653 OJ: Multiplying Rational Expressions ontaining Polynomials NT: c ST: TOP: 10-4 Multiplying and ividing Rational Expressions 38. NS: 1 n n 7 7n 1 n 7n Write as multiplication by the reciprocal. n 7 1(7n) Multiply the numerators and the denominators. n( n 7) 7 ivide out common factors. Simplify. n 7 ivide out common factors, and simplify. Write as multiplication by the reciprocal first. First, write as multiplication by the reciprocal. Then, multiply the numerators and the denominators. PTS: 1 IF: asic REF: Page 654 OJ: ividing by Rational Expressions and Polynomials NT: c ST: TOP: 10-4 Multiplying and ividing Rational Expressions 20

29 I: 39. NS: 3x + 8 x x x x + 12 x 2 16 x (x + 4) (x + 4)(x 4) 3 x 4 ombine like terms in the numerator. Factor. ivide out common factors. Simplify. ivide out only common factors. Factor the denominator and continue simplifying your answer. The denominators are the same, so keep the common denominator. PTS: 1 IF: verage REF: Page 659 OJ: dding Rational Expressions with Like enominators NT: c ST: TOP: 10-5 dding and Subtracting Rational Expressions 40. NS: 3y 9y + 3y Identify the L, 18y y 3y 2 9y y y 18y y 6y 18y 2 + 3y 2 18y 2 Rewrite each fraction with a denominator of 18y y + 3y 18y 2 3y(2 + y) 3y(6y) 2 + y 6y dd. 6y and 3y 2 are not like terms, so they cannot be combined. Factor and divide out common factors. Simplify. To simplify the numerator, factor out 3y. First find a common denominator. Then add the numerators. To simplify the numerator, factor out 3y. PTS: 1 IF: asic REF: Page 661 OJ: dding and Subtracting with Unlike enominators NT: c ST: TOP: 10-5 dding and Subtracting Rational Expressions 21

30 I: 41. NS: (12x 4 18x x 2 ) 6x 3 12x 4 18x x 2 6x 3 12x 4 18x x 2 6x 3 6x 3 6x 3 2x 4 3x 3 + 6x 2 x 3 x 3 x 3 2x x Rewrite as a rational expression. ivide each term in the numerator by the denominator. Simplify by dividing out common factors. Simplify by using powers of exponents. Write the division of each term by the divisor as a quotient. Simplify by dividing out common factors and using the rule of exponents. Write the division of each term by the divisor as a quotient. Simplify by dividing out common factors and using the rule of exponents. Write the division of each term by the divisor as a quotient. Simplify by dividing out common factors and using the rule of exponents. PTS: 1 IF: verage REF: Page 667 OJ: ividing a Polynomial by a Monomial ST: TOP: 10-6 ividing Polynomials 42. NS: 2x x 3 x 3 2x 2 + 3x 15 NT: c Write the problem in long division form. There will be a remainder when you finish with the long division. Write the remainder as a rational expression using the divisor as the denominator. Use long division. Write the remainder as a rational expression using the divisor as a denominator. Use long division. PTS: 1 IF: verage REF: Page 670 OJ: Long ivision with a Remainder ST: TOP: 10-6 ividing Polynomials NT: c 22

31 I: 43. NS: 4 m 3 3 5m 4( 5m) 3( m 3) Use cross products. 20m 3m 9 Multiply. istribute 3 on the right side.. 17m 9 Subtract 3m from both sides. m 9 17 ivide both sides by 17. First compute the cross products. Then solve for the variable. First compute the cross products. Then solve for the variable. First compute the cross products. Then solve for the variable. PTS: 1 IF: verage REF: Page 674 OJ: Solving Rational Equations by Using ross Products NT: a ST: TOP: 10-7 Solving Rational Equations 44. NS: Multiply both sides of the equation by the L: 3z(z 21). ivide out common factors, then distribute. This gives the equation: 9z z 84 30z ombine like terms and simplify. Then solve for z. When solving for the variable, perform the same operations on both sides of the equation. Find the L of all the rational expressions in the equation. Keep track of signs when multiplying by the L and solving for the variable. PTS: 1 IF: verage REF: Page 674 OJ: Solving Rational Equations by Using the L ST: TOP: 10-7 Solving Rational Equations NT: a 23

32 I: 45. NS: Factor perfect squares out of the radicand. Use the Product Property of Square Roots to take the square root of each factor separately. Simplify. First, factor perfect squares out of the radicand. Then, take the square roots of the perfect squares. Take the square root of the factor when removing the radical. First, factor perfect squares out of the radicand. Then, take the square roots of the perfect squares. PTS: 1 IF: verage REF: Page 706 OJ: Using the Product Property of Square Roots ST: 12.0 TOP: 11-2 Radical Expressions 46. NS: b 9 16b NT: c b 4 4 b 8 16 Simplify the radicand. Use the Quotient Property of Square Roots. always indicates a nonnegative square root. Simplify the numerator. Find the square root of the radicand (the expression under the radical sign). The radical sign always indicates a nonnegative square root. Find the square root of the expression. PTS: 1 IF: verage REF: Page 706 OJ: Using the Quotient Property of Square Roots ST: 12.0 TOP: 11-2 Radical Expressions NT: c 24

33 I: 47. NS: (5) Use the Quotient Property. Find perfect square factors if possible. Write 80 as 16(5). Use the Product Property. Simplify. Factors that aren't perfect squares remain beneath the radical sign. Only factors that aren't perfect squares remain beneath the radical sign. heck your calculations. factor from the numerator is missing. PTS: 1 IF: asic REF: Page 707 OJ: Using the Product and Quotient Properties Together NT: c ST: 12.0 TOP: 11-2 Radical Expressions 48. NS: 75b b 2 27b 25(3b) + 4 4(3b) 2 9(3b) Factor the radicands using perfect-square factors. 5 3b + 4(2) 3b 2(3) 3b Use the Product Property of Square Roots. 5 3b + 8 3b 6 3b 7 3b ombine like radicals. The answer should have the radical sign with at least the variable as the radicand. You cannot square the radical expression without changing its value. Factor the radicands using perfect-square factors. Use the Product Property of Square Roots. PTS: 1 IF: verage REF: Page 712 OJ: Simplifying efore dding or Subtracting NT: c ST: 12.0 TOP: 11-3 dding and Subtracting Radical Expressions 25

34 I: 49. NS: 25h 10h 250h 2 Product Property of Square Roots. 2 Factor 250 using a perfect-square factor. 25( 10)h h 2 Product Property of Square Roots. Ê ˆ 5 10 Simplify. Ë Á h Take the square root of the perfect-square factor. Use the Product Property of Square Roots. Then factor the number using a perfect-square factor. Factor the number using a perfect-square factor. PTS: 1 IF: dvanced REF: Page 716 OJ: Multiplying Square Roots NT: c ST: 12.0 TOP: 11-4 Multiplying and ividing Radical Expressions 50. NS: Ê 5 Ë Á ˆ 5 5 Multiply by a form of 1 to get a perfect-square radicand in the denominator. Simplify the denominator. First, multiply by a form of 1 to get a perfect-square radicand in the denominator. Then, simplify the denominator. Rationalize the denominator by finding the appropriate form of 1 to multiply by. quotient with a square root in the denominator is not simplified. PTS: 1 IF: asic REF: Page 718 OJ: Rationalizing the enominator NT: c ST: 12.0 TOP: 11-4 Multiplying and ividing Radical Expressions 26

35 I: 51. NS: b 18 b b 324 Square both sides. The radical cannot be negative. Square both sides, not multiply. Square both sides to isolate the variable. PTS: 1 IF: asic REF: Page 722 OJ: Solving Simple Radical Equations ST: 12.0 TOP: 11-5 Solving Radical Equations 52. NS: Isolate the radical on one side of the equation. Square both sides. Solve for y. heck your answer in the original equation to determine if a solution is extraneous. heck your answer in the original equation. Square both sides of the equation. Square both sides of the equation. PTS: 1 IF: verage REF: Page 725 OJ: Extraneous Solutions ST: TOP: 11-5 Solving Radical Equations 27