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1 Name: Class: Date: ID: A Chapter 8: Factoring Polynomials--Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the prime factorization of 70. a. 2 5 b c d Find the GCF of 48 and 72. a. 72 b. 24 c. 48 d Find the GCF of2m 5 and 32m 4 a. 32m 4 c. 2m b. m 4 d. 2m 4 4. The brass section in a marching band has 24 trombone players and 56 trumpet players. The band director wants to arrange them into equal rows that have as many musicians as possible. However, all of the musicians in each row must play the same instrument. How many rows will there be? a. There will be 8 rows. c. There will be 20 rows. b. There will be 10 rows. d. There will be 3 rows. 5. The amount of paint needed to cover a wall is proportional to its area. The wall is rectangular and has an area of 4Z2 + 2z square meters. Factor this polynomial to find possible expressions wall. (Assume the factors are polynomials.) a. 2z(2z + 1); possible dimensions: 2z meters by (2z + 1) meters b. 2z2 (2z + 1); possible dimensions: 2Z2meters by (2z + 1) meters c. 2z( 4z + 2); possible dimensions: 2z meters by (4z + 2) meters d. 2(2z+z); possible dimensions: 2 meters by (2z + z) meters 6. Factor 5(x - 2) - 9x(x - 2). a. -45x(x - 2) b. (x - 2)(9x - 5) 7. Factor X2+ 101x a. (x + 101)(x + 100) b. (x + 2)(x + 50) 8. Factor the trinomial a' + 14a a. (a + 14)(a + 1) b. (a+1)(a+48) 9. Factor the trinomial r 2 + r a. (r-4)(r+5) b. (r-5)(r-4) c. d. c. d. (5-9x)(x - 2)(x - 2) (x - 2)(5-9x) (x + 5)(x + 20) (x + l)(x + 100) c. (a+6)(a + 8) d. (a - 8)(a - 6) c. (r + 1)(r - 20) d. (r - l)(r - 20) for the length and width of the

2 Name: ID: A 10. Factor X2 + 20x Check that the original polynomial and the factored form have the same values for x = 0, 1,2,3, and 4. a. (x + 20)(x + 36) c. (x + 4)(x + 9) b. (x + lo)(x + 10) d. (x + 2)(x + 18) 11. Factor 3x 2 + 2x - 8. a. (x - 2)(3x + 4) b. (x + 2)(3x + 4) 12. Factor 2x2 + 7x + 6. a. (x+ 3)(2x +2) c. (x+2)(x+3) b. (x + 2)(2x - 3) d. (x + 2)(2x + 3) 13. Find all possible values of b such that 4X2 + bx + 3 can be factored. a. 7, 8 c. 7 b. 7,8,13 d. 8, Determine whether 16x 2-24x + 9 is a perfect square. If so, factor it. If not, explain why. a. Yes, 16x 2-24x + 9 is a perfect square. (l6x - 9) 2 b. No, 16x 2-24x + 9 is not a perfect square. 16x 2 and 9 are perfect squares, but 24x is not a perfect square. So 16x 2-24x + 9 is not a perfect square. c. Yes, 16x 2-24x + 9 is a perfect square. (4x + 3) 2 d. Yes, 16x 2-24x + 9 is a perfect square. (4x - 3) TeJI whether the polynomial 6y2(y2 + 6y + 9) is completely factored. If not, factor it. a. Yes. c. No; 6y4 + 36y3 + 54y2. b. No; 6y2(y+3)(y-3). d. No; 6y2(y+3) Factor 27x 2 z + 36xz + 12z completely. a. z(3x + 12) 2 c. 12z(2x 2 + 3x + 1) b. 3z(3x + 2) 2 d. 3z(3x + 2) (3x - 2) 17. Write the polynomial in standard form. Then name the polynomial based on its degree and number of terms. 2 - l lx? - 8x + 6x 2 c. (x - 2)(3x - 4) d. (x + 2)(3x - 4) a. -5X2-8x + 2; quadratic trinomial b. 5x 2-8x - 2; quadratic trinomial c. -6x 2-8x - 2; cubic polynomial d. 6x 2-8x + 2; cubic trinomial Find the degree of the monomial m 6 n 5 a. 5 b. 11 c. 6 d Match the expression with its name. 6x 3-9x + 3 a. cubic trinomial c. fourth-degree monomial b. quadratic binomial d. not a polynomial 2

3 Name: ID: A 20. Write the perimeter of the figure. 3x+2 6x 5x not to scale a. 9x+7x h. llx + 3x + 2 c. 14x+ 2 d. 14x Simplify the product n(n 2 + 3n + 4) a. 2n 3 + 6n 2 + 8n c. 2n 3 + 6n + 8 h. 2n 3 + 3n + 4 d. n 2 + 5n + 4 Factor the polynomial x 3 + 4X2 + &x a. 2x(x 2 + 2x + 4) c. x(2x 2 + 4x + 8) h. 2x(x + 2)(x + 4) d. 2x 3 + 4X2 + 8x 23. Find the GCF of the terms of the polynomial. 8x x 3 a. x 3 h. 8x 3 c. 8x 3 d. 8x 6 Simplify the product using FOIL. 24. (3x - 7)(3x - 5) a. 9x 2 + 6x + 35 c. 9x2-36x- 35 h. 9x x + 35 d. 9x 2-36x + 35 Find the square. 25. (2x- 6)2 a. 4X2-24x + 36 c. 4X h. 4X2-8x + 36 d. 4X2 -l2x

4 Name: : A 26. Find the area of the UNSHADED region. Write your answer in standard form. x x+5 a. _2X2 + lox + 25 c. 10x+ 25 h. X2+ 8x+ 25 d. X2+ lox + 25 Factor the expression. 27. w w + 77 a. (w - 7)(w + 11) c. (w + 7)(w + 11) h. (w -7)(w -11) d. (w + l)(w + 77) 28. tf2+ 10d+9 a. (d+ 9)(d-1) c. (d-9)(d-1) h. (d- 9)(d+ 1) d. (d+ 9)(d+ 1) 29. x2-x-42 a. (x-7)(x + 6) c. (x + 7)(x - 6) h. (x + 7)(x + 6) d. (x -7)(x - 6) 30. d 2-14d + 49 a. (d + 7)2 c. (d - 7)(d + 7) h. (d - 7)2 d. (d - 49)(d - 1) 4

5 Chapter 8: Factoring Polynomials--Review Answer Section MULTIPLE CHOICE 1. ANS: B Choose a prime factor of 70 to begin. Keep dividing by prime factors until the quotient is 1. A Check your answer by multiplying the factors. B Correct! C Check your answer by multiplying the factors. 0 Check your answer by multiplying the factors. PTS: 1 DIF: Basic REF: Page 524 OBJ: Writing Prime Factorization NAT: b TOP: 8-1 Factors and Greatest Common Factors KEY: prime factorization I exponents 2. ANS: B List all the factors of 48 and 72. Choose the greatest factor that is in both lists. A Check to see that this is a factor of both numbers. If so, check that it is the GREATEST common factor. B Correct! C Check that this is a factor of the greater number. 0 This is the least common multiple (LCM) of the two numbers. Find the greatest common factor (GCF). PTS: 1 DIF: Basic REF: Page 525 OBJ: Finding the GCF of Numbers NAT: b TOP: 8-1 Factors and Greatest Common Factors KEY: common factor I factor I GCF I greatest common factor 3. ANS: D First, write the prime factorization of each coefficient and write the powers as products. Then, find the product of the common factors. A Find the GCF of the coefficients. B Find the product of all the common factors. C There are more factors in common. 0 Correct! PTS: 1 DIF: Average REF: Page 525 OBJ: Finding the GCF of Monomials TOP: 8-1 Factors and Greatest Common Factors NAT: b 1

6 4. ANS: B First, fmd the GCF of the number of trombone players and the number oftrurnpet of musicians in each row. players. This is the number Since the GCF of24 and 56 is 8, there are 8 musicians in each row. Divide to find the number ofrows of trombone players: = 3. Divide to find the number ofrows of trombone players: = 7. The total number ofrows = = 10. A This is the number in each row. You need to fmd the number of rows. B Correct! C Check your division. Use the GCF to fmd the number of musicians in each row, and then divide to find the total number of rows. 0 This is the number ofrows of trombone players. You need to fmd the total number of rows. PTS: 1 DIP: Average REF: Page 526 OBJ: Application NAT: b STA: A.4.A TOP: 8-1 Factors and Greatest Cornmon Factors KEY: GCF I greatest cornmon factor I factor 5. ANS: A A = 4Z2 +2z A = 2z(2z) + 2z(1) A = 2z(2z + 1) The GCF is 2z. Write each term as a product with the GCF as a factor. Use the Distributive Property to factor out the GCF. A Correct! B Check your answer by multiplying the factors. C Check your answer by multiplying the factors. 0 Check your answer by multiplying the factors. PTS: 1 NAT: d DIP: STA: Average A.4.A REF: Page 532 OBJ: Application TOP: 8-2 Factoring by GCF 2

7 6. ANS: D 5(x -2) - 9x(x- 2) The terms have a common binomial factor of (x - 2). Factor out (x - 2). (x - 2)(5-9x) A B C D Check the signs. Check the order of the second factor terms. The common factor should appear only once. Correct! PTS: 1 DIF: Basic REF: Page 533 OBJ: Factoring Out a Common Binomial Factor NAT: d TOP: 8-2 Factoring by GCF 7. ANS: D The factors of X2 + 10lx are binomials. Use FOIL as a guide (First, Outer, Inner, Last) to fmd the binomial factors. The first terms multiply to X2. (x +?)(x +?) = X2 The last terms multiply to 100. Check each pair of factors of 100 to find the inner and outer products that add to the middle term 10lx. (x + l)(x + 100) (x + 2)(x + 50) (x + 4)(x + 25) (x + 5)(x + 20) (x + 10)(x + 10) X x X2 + 52x X2 + 29x x 2 +25x+100 X2 + 20x The only factors of that 100 produce the correct middle term are 1 and 100. X x = (x + l)(x + 100) A Use factors of the constant term. The coefftcient of the x term is the sum of the inner and outer products. B Check the factors. The coefficient of the x term is the sum of the inner and outer products. C Check the factors. The coefftcient of the x term is the sum of the inner and outer products. D Correct! PTS: 1 DIF: Average REF: Page 540 OBJ: Factoring Trinomials by Guess and Check TOP: 8-3 Factoring xl\2 + bx + C NAT: d 3

8 8. ANS: C a 2 +14a+48 (a+?)(a+?) (a+6)(a+8) Look for the factors of 48 whose sum is 14. The factors are 6 and 8. A Look for factors whose product is the trinomial's last term. S Use the FOIL method to check your answer. C Correct! 0 Use the FOIL method to check your answer. PTS: 1 DIF: Basic REF: Page 541 OBJ: Factoring x/\2 + bx + c When c is Positive TOP: 8-3 Factoring x/\2 + bx + C 9. ANS: A r 2 +r-20 (r+?)(r+?) (r-4)(r+ 5) NAT: d Look for the factors of -20 whose sum is 1. The factors are -4 and 5. A Correct! S Use the FOIL method to check your answer. C Use the FOIL method to check your answer. 0 Check the signs. PTS: 1 DIF: Basic REF: Page 542 OBJ: Factoring x/\2 + bx + c When c is Negative TOP: 8-3 Factoring x/\2 + bx + C NAT: d 4

9 10. ANS: D X2 +20x+36 (x+?)(x +?) (x + 2)(x + 18) Look for factors of36 whose sum is 20. The factors needed are 2 and 18. EvaIuaet th e ongma. I potynomia.i x-? + 20x + 36 and the factored form (x + 2)(x + 18) for x = 0, 1,2,3, and 4. x X2 + 20x+ 36 x (x + 2)(x + 18) 0 ()2+ 20(0) + 36 = (1)+36 = (2) + 36 = (3) + 36 = (4) + 36 = (0 + 2)(0 + 18) = 36 1 (1 + 2)(1 + 18) = 57 2 (2 + 2)(2 + 18) = 80 3 (3 + 2)(3 + 18) = (4 + 2)(4 + 18) = 132 The original polynomial and the factored form have the same value for the given values of x. This is a way to verify that X2 + 20x + 36 = (x + 2)(x + 18). A Multiply the binomials to check your answer. S Look for factors of36 whose sum is 20. C Look for factors of36 whose sum is 20. D Correct! PTS: 1 DIF: Average REF: Page 543 OBJ: Evaluating Polynomials NAT: d TOP: 8-3 Factoring x/\2 + bx + c 11. ANS: D Try factors of 3 for the coefficients and factors of -8 for the constant tenus. The combination that works is: (x + 2)(3x - 4) = 3x 2-4x + 6x - 8 = 3x 2 + 2x - 8 A S C D Multiply the factors to check your answer. Check the signs. Multiply the factors to check your answer. Correct! PTS: 1 DIF: Basic REF: Page 548 OBJ: Factoring ax/\2 + bx + c by Guess and Check NAT: d TOP: 8-4 Factoring ax/\2 + bx + c KEY: factor I trinomial I guess and check 5

10 12. ANS: D Since a = 2, the coefficients of the First terms must be factors of2. Since c = 6, the Last terms must be factors of 6. Since b = 7, the Outer and Inner products must add up to 7. The sum of the products of the outer and inner terms should be 7. It may be helpful to make a table to check all the factors of 2 and all the factors of 6. Then check the products of the outer and inner terms to see ifthe sum is 7. A You reversed the second terms in the parentheses. B When b is negative, the factors of c are both negative. When b is positive, the factors of c are both positive. C The coefficient of the x-term in the second binomial cannot be 1. Check your answer. 0 Correct! PTS: 1 DIF: Basic REF: Page 549 OBJ: Factoring ax/\2 + bx + c When c is Positive TOP: 8-4 Factoring ax/\2 + bx + C 13. ANS: B Find all possible combinations of the factors of 4X2 and 3. Factors of 4X2 Factors of3 NAT: d 4x and x 3 and 1 (4x + 3)(x + 1) b=4+3=7 4x and x 1 and 3 (4x+ 1)(x+3) b = = 13 2x and2x 3 and 1 (2x + 3)(2x + 1) b=2+6=8 The possible values of bare 7,8, and 13. A There are more possible values. B Correct! C There are more possible values. 0 Find all possible combinations of the factors of 4x/\2 and 3. PTS: 1 DIF: Advanced TOP: 8-4 Factoring ax/\2 + bx + C 6

11 14. ANS: D 16x 2-24x + 9 is a perfect square if: * The first and last terms are perfect squares. * The middle term is two times the square root of the first term and the square root of the last term. A B C D Factor correctly. Find the product of the square roots of the first and last terms. Then multiply that product by 2 to see if it is equal to the middle term. Check the signs. Correct! PTS: 1 DIF: Average REF: Page 558 OBJ: Recognizing and Factoring the Perfect-Square Trinomials NAT: d TOP: 8-5 Factoring Special Products 15. ANS: D The factor (y2 + 6y + 9) is a perfect square trinomial and can be factored. A B C D There is a perfect square trinomial that can be factored. Multiply the factors to check your answer. Multiply the factors to check your answer. Correct! PTS: 1 DIF: Advanced REF: Page 566 OBJ: Determining Whether a Polynomial is Completely Factored NAT: d TOP: 8-6 Choosing a Factoring Method 16. ANS: B Factor out the GCF, 3z. The remaining polynomial, 9x x + 4, is a perfect square trinomial that can be factored. A B C D Check the GCF. Correct! Check the GCF. After you factor out the GCF, the remaining polynomial is a perfect square trinomial that can be factored. PTS: 1 DIF: Advanced REF: Page 567 OBJ: Factoring by GCF and Recognizing Patterns NAT: d TOP: 8-6 Choosing a Factoring Method 17. ANS: A PTS: 1 DIF: L3 REF: 9-1 Adding and Subtracting Polynomials OBJ: Describing Polynomials NAT: NAEP 2005 A3b I ADP J.1.3 STA: TX TEKS A.3A I TX TEKS AAA TOP: 9-1 Example 2 KEY: monomial I degree of a monomial I polynomial I degree of a polynomial polynomial I trinomial I classifying polynomials I multi-part question I standard form of a 7

13 28. ANS: D PTS: 1 DIF: L3 REF: 9-5 Factoring Trinomials ofthe Type x/\2 + bx + c OBJ: Factoring Trinomials NAT: NAEP 2005 A3c I ADP J.1.4 STA: TX TEKS A.4A TOP: 9-5 Example 1 KEY: polynomial I factoring trinomials 29. ANS: A PTS: 1 DIF: L3 REF: 9-5 Factoring Trinomials of the Type x/\2 + bx + c OBJ: Factoring Trinomials NAT: NAEP 2005 A3c I ADP J.1.4 STA: TX TEKS A.4A TOP: 9-5 Example 3 KEY: polynomial I factoring trinomials 30. ANS: B PTS: 1 DIF: L2 REF: 9-7 Factoring Special Cases OBJ: Factoring Perfect-Square Trinomials NAT: ADP J.1.4 STA: TX TEKS A.4A TOP: 9-7 Example 1 KEY: polynomial I factoring trinomials I perfect-square trinomial 9

14 Chapter 8: Factoring Polynomials--Review [Answer Strip] ID:A D 10. C 20. C 26. B 1. D 11. D 12. B 2. B 13. D 3. D 14. A 21. B 4. C 27. A 5. D 15. A 22. D 28. B 16. B 23. A 29. D 6. D 7. A 17. D 24. B 30. C 8. B 18. A 9. A 19. A 25.

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