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Chapter 12 Skills Practice 617
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Lesson 12.1 Skills Practice
Name Date
Controlling the PopulationAdding and Subtracting Polynomials
Vocabulary
Match each definition with its corresponding term.
1. polynomial a. a polynomial with only 1 term
2. term b. the degree of the term with the greatest exponent
3. coefficient c. a mathematical expression involving the sum of powers in one or more variables multiplied by coefficients
4. monomial d. a polynomial with exactly 3 terms
5. binomial e. any number being multiplied by a power within a polynomial expression
6. trinomial f. each product in a polynomial expression
7. degree of a term g. a polynomial with exactly 2 terms
8. degree of a polynomial h. the exponent of a term in a polynomial
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618 Chapter 12 Skills Practice
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Lesson 12.1 Skills Practice page 2
Problem Set
Identify the terms and coefficients in each expression.
1. 5x 1 8
The terms are 5x and 8. The coefficients are 5 and 8.
2. 2m3
3. x2 2 4x 4. 23w4 1 w2 2 9
5. 218 6. 10 2 3x3 2 6x
Determine whether each expression is a polynomial. If the expression is not a polynomial, explain why it is not.
7. 9 1 12x
The expression is a polynomial.
8. 6m1 __ 2
9. 3 __ x 2 8x 10. 22w3 1 w2 2 5
11. 22.5m 12. x__7 1 10
13. 3 x 1 12 14. 4 __5 m 2 1 __
5
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Chapter 12 Skills Practice 619
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Lesson 12.1 Skills Practice page 3
Name Date
Determine whether each polynomial is a monomial, binomial, or trinomial. State the degree of the polynomial.
15. 8x 1 3
The polynomial is a binomial with a degree of 1.
16. 5m 2
17. x2 2 7x 18. 29n4 1 6n2 2 1
19. 212 20. 4 2 10x3 1 8x
Write each polynomial in standard form. Classify the polynomial by its number of terms and by its degree.
21. 2x 1 6x2
6x2 1 2x
The polynomial is a binomial with a degree of 2.
22. 29m2 1 4m3
23. 10 2 5x 24. 7x 2 3 1 12x2
25. 15 1 4w 2 w3 26. 5x2 2 15 1 20x
27. 21 2 p4 28. 26t2 1 4t 1 3t3
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620 Chapter 12 Skills Practice
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Lesson 12.1 Skills Practice page 4
29. 218a3 1 54a 2 22a2 30. x3 2 x2 2 x5
Simplify each expression.
31. (5x 2 8) 1 (7x 110)
5x 2 8 1 7x 1 10
(5x 1 7x) 1 (28 1 10)
12x 1 2
32. (4m2 1 9m) 2 (2m2 1 6)
33. (2x2 1 5x 2 12) 1 (2x2 2 6) 34. (10t2 2 3t 1 9) 2 (6t2 2 7t)
35. (25w2 1 3w 2 8) 1 (15w2 2 4w 1 11) 36. (3x3 1 10x 2 1) 2 (5x2 1 10x 2 9)
37. (2a2 1 2a 2 8) 1 (2a2 2 9a 1 15) 38. (14p4 1 7p2) 1 (8p3 1 7p2 2 p)
39. (3x4 1 3 x 2 2 3) 2 (6 x 5 2 9 x 3 1 2) 40. (27m3 2 m2 2 m) 2 (210m3 2 m 2 1)
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Chapter 12 Skills Practice 621
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Lesson 12.1 Skills Practice page 5
Name Date
The graphs of the functions f(x) 5 2x 1 1, g(x) 5 x 2 1 x 2 3 , and h(x) 5 f(x) 1 g(x) are shown. Evaluate the function h(x) for each given value of x. Use the graph of h(x) to verify your answer.
24 23 22 21 0 1
2
22
24
26
28
4
6
8
2 3 4x
yg(x)
h(x)
f(x)
41. Evaluate h(x) at x 5 2.
h(x) 5 f(x) 1 g(x)
5 2x 1 1 1 x 2 1 x 2 3
5 x 2 1 3x 2 2
h(2) 5 (2 ) 2 1 3(2) 2 2
5 4 1 6 2 2
5 8
42. Evaluate h(x) at x 5 24.
43. Evaluate h(x) at x 5 0. 44. Evaluate h(x) at x 5 1.
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622 Chapter 12 Skills Practice
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Lesson 12.1 Skills Practice page 6
45. Evaluate h(x) at x 5 22. 46. Evaluate h(x) at x 5 21.5.
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Chapter 12 Skills Practice 623
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Lesson 12.2 Skills Practice
Name Date
They’re Multiplying—Like Polynomials!Multiplying Polynomials
Problem Set
Determine the product of the binomials using algebra tiles.
1. x 1 1 and x 1 1
x xx2
1
11
x
x
(x 1 1) (x 1 1) 5 x 2 1 2x 1 1
2. x 1 1 and x 1 4
3. x 1 2 and x 1 2
4. x 1 3 and x 1 3
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624 Chapter 12 Skills Practice
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Lesson 12.2 Skills Practice page 2
5. 2x 1 1 and x 1 3
6. 2x 1 3 and x 1 2
Determine the product of the binomials using multiplication tables.
7. 3x 1 4 and 2x 1 2
? 2x 2
3x 6x 2 6x
4 8x 8
(3x 1 4)(2x 1 2) 5 6 x 2 1 6x 1 8x 1 8
5 6 x 2 1 14x 1 8
8. 5m 1 3 and 4m 1 6
9. 6t 1 5 and 7t 2 5
10. 4x 1 2 and 4x 2 2
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Chapter 12 Skills Practice 625
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Lesson 12.2 Skills Practice page 3
Name Date
11. 10w 2 1 and 9w 1 8
12. y 1 12 and 5y 1 15
Determine the product of the polynomials using the Distributive Property.
13. 2x(x 1 6)
2x(x 1 6) 5 2x(x) 1 2x(6)
5 2 x 2 1 12x
14. 4 x 2 (x 1 2)
15. 7x(x 2 5)
16. (2x 1 1)(x 1 8)
17. (x 1 3)( x 2 2 1)
18. (4x 1 4)(5x 2 5)
19. 3x( x 2 1 5x 2 1)
20. 9x(3 x 2 2 4x 1 2)
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626 Chapter 12 Skills Practice
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Lesson 12.2 Skills Practice page 4
21. (x 1 2)( x 2 1 6x 2 1)
22. (x 2 4)( x 2 1 2x 2 3)
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Chapter 12 Skills Practice 627
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What Factored Into It?Factoring Polynomials
Vocabulary
State the given property.
1. Symmetric Property of Equality
Problem Set
Factor out the greatest common factor of each polynomial, if possible.
1. x 2 1 9x
x(x 1 9)
2. m 2 2 4m
3. 5 x 2 1 20x 2 15 4. 24 w 2 2 16
5. y 3 2 7y 6. 2 x 3 1 10 x 2
7. 3w 1 10 8. 20 x 3 1 16 x 2 1 8x
9. 7 m 3 2 21 10. 15 x 3 1 4
Lesson 12.3 Skills Practice
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628 Chapter 12 Skills Practice
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Lesson 12.3 Skills Practice page 2
Factor each trinomial using an area model.
11. x 2 1 4x 1 3
xxx2
1 1
x
1x
1 1 1x
x 1 3
x 1 1x
1
x 2 1 4x 1 3 5 (x 1 1)(x 1 3)
12. x 2 1 5x 1 6
13. x 2 2 x 2 6
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Chapter 12 Skills Practice 629
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Lesson 12.3 Skills Practice page 3
Name Date
14. x 2 2 x 2 12
15. x 2 1 7x 1 10
16. x 2 1 3x 2 4
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630 Chapter 12 Skills Practice
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Lesson 12.3 Skills Practice page 4
Factor each trinomial completely using multiplication tables. If possible, factor out the greatest common factor first.
17. x 2 2 2x 2 8
? x 2
x x 2 2x
24 24x 28
x 2 2 2x 2 8 5 (x 2 4)(x 1 2)
18. y 2 1 13y 1 42
19. m 2 1 6m 2 7
20. x 2 2 9x 1 18
21. 4 w 2 1 12w 2 40
22. 2 t 3 2 14 t 2 1 24t
23. 3 m 3 1 36 m 2 1 60m
24. 2 x 2 2 8x 2 42
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Chapter 12 Skills Practice 631
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Lesson 12.3 Skills Practice page 5
Name Date
Factor each polynomial using the trial and error method. If possible, factor out the greatest common factor first.
25. x 2 1 11x 1 10
The factors of the constant term, 10, are:
21, 210 1, 10
22, 25 2, 5
x 2 1 11x 1 10 5 (x 1 1)(x 1 10)
26. w 2 1 6w 2 16
27. m 2 1 2m 2 35 28. x 2 1 4x 2 12
29. 3 n 2 2 27n 1 60 30. 2 x 2 1 22x 1 60
Factor each polynomial.
31. x 2 1 11x 1 28 5 (x 1 4)(x 1 7)
x 2 2 11x 1 28 5 (x 2 4)(x 2 7)
x 2 1 3x 2 28 5 (x 2 4)(x 1 7)
x 2 2 3x 1 28 5 (x 1 4)(x 2 7)
32. x 2 1 10x 1 9 5
x 2 2 10x 1 9 5
x 2 1 8x 2 9 5
x 2 2 8x 2 9 5
33. x 2 1 12x 1 27 5
x 2 2 12x 1 27 5
x 2 1 6x 2 27 5
x 2 2 6x 2 27 5
34. x 2 1 13x 1 40 5
x 2 2 13x 1 40 5
x 2 1 3x 2 40 5
x 2 2 3x 2 40 5
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632 Chapter 12 Skills Practice
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Lesson 12.3 Skills Practice page 6
35. x 2 1 12x 1 11 5
x 2 2 12x 1 11 5
x 2 1 10x 2 11 5
x 2 2 10x 2 11 5
36. x 2 1 13x 1 36 5
x 2 2 13x 1 36 5
x 2 1 5x 2 36 5
x 2 2 5x 2 36 5
Factor each polynomial completely. If possible, factor out the greatest common factor first.
37. x 2 1 4x 1 4
? x 2
x x 2 2x
2 2x 4
x 2 1 4x 1 4 5 (x 1 2)(x 1 2)
38. x 2 2 10x 1 25
39. 232 2 12m 2 m 2
40. 45 1 4w 2 w 2
41. 5 x 2 1 10x 2 15
42. 4 x 2 1 32x 1 64
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Chapter 12 Skills Practice 633
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Zeroing InSolving Quadratics by Factoring
Vocabulary
Complete the definition of the Zero Product Property.
1. The Zero Product Property states that if the product of two or more factors is equal to , then at least one factor must be equal to .
If ab 5 0, then or .
This property is also known as the .
Define the term in your own words.
2. roots
Problem Set
Factor and solve each quadratic equation. Check your answer.
1. x 2 1 5x 1 6 5 0
x 2 1 5x 1 6 5 0 Check:
(x 1 3)(x 1 2) 5 0 (2 3) 2 1 5(23) 1 6 5 0 (2 2) 2 1 5(22) 1 6 5 0
x 1 3 5 0 or x 1 2 5 0 9 2 15 1 6 5 0 4 2 10 1 6 5 0
x 5 23 or x 5 22 0 5 0 0 5 0
The roots are 23 and 22.
2. x 2 2 3x 2 4 5 0
Lesson 12.4 Skills Practice
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634 Chapter 12 Skills Practice
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Lesson 12.4 Skills Practice page 2
3. m 2 1 2m 2 35 5 0
4. 2 x 2 2 4x 1 12 5 0
5. x 2 1 8x 5 0
6. w 2 1 50 5 215w
7. 2 t 2 1 12t 5 32
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Chapter 12 Skills Practice 635
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8. x 2 1 2x 1 2 5 0
9. 2 t 2 1 t 2 3 5 0
10. w 2 1 5w 2 32 5 2w 2 4
Lesson 12.4 Skills Practice page 3
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636 Chapter 12 Skills Practice
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Lesson 12.4 Skills Practice page 4
Determine the zeros of each quadratic function, if possible. Check your answer.
11. f(x) 5 x 2 2 5x
f(x) 5 x 2 2 5x Check:
0 5 x 2 2 5x (0 ) 2 2 5(0) 0 0 (5 ) 2 2 5(5) 0 0
0 5 x(x 2 5) 0 2 0 0 0 25 2 25 0 0
x 5 0 or x 2 5 5 0 0 5 0 0 5 0
x 5 0 or x 5 5
The zeros are 0 and 5.
12. f(x) 5 3 x 2 1 6x
13. f(x) 5 x 2 1 11x 1 30
14. f(x) 5 x 2 2 9x 2 36
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Chapter 12 Skills Practice 637
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Lesson 12.4 Skills Practice page 5
Name Date
15. f(x) 5 2 x 2 1 9x 1 10
16. f(x) 5 x 2 1 5x 1 14
17. f(x) 5 3 x 2 1 3x 2 6
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638 Chapter 12 Skills Practice
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Lesson 12.4 Skills Practice page 6
18. f(x) 5 1 __ 2 x 2 2 3 __
4 x
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Chapter 12 Skills Practice 639
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What Makes You So Special?Special Products
Vocabulary
Give an example of each term. Then, factor the expression.
1. perfect square trinomial
2. difference of two squares
3. sum of two cubes
4. difference of two cubes
Problem Set
Factor each binomial completely.
1. x 2 2 25
x 2 2 25 5 (x 1 5)(x 2 5)
2. x 3 2 64
3. x 3 1 27 4. m 2 2 100
Lesson 12.5 Skills Practice
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Lesson 12.5 Skills Practice page 2
5. 5 x 3 1 40 6. t 3 2 125
7. 8 a 3 2 27 8. x 8 2 y 8
Factor the trinomial completely.
9. x 2 1 16x 1 64
x 2 1 16x 1 64 5 (x 1 8)(x 1 8)
10. k 2 2 20k 1 100
11. 2 x 2 2 28x 1 98 12. 5 x 2 1 10x 1 5
13. z 3 1 18 z 2 1 81z 14. 3x 3 2 30 x 2 1 75x
Determine the root(s) of each quadratic equation. Check your answer(s).
15. x 2 2 100 5 0
x 2 2 100 5 0 Check:
(x 1 10)(x 2 10) 5 0 (2 10) 2 2 100 0 0 ( 10) 2 2 100 0 0
x 1 10 5 0 or x 2 10 5 0 100 2 100 0 0 100 2 100 0 0
x 5 210 or x 5 10 0 5 0 0 5 0
The roots are 210 and 10.
16. m 2 2 16m 1 64 5 0
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Chapter 12 Skills Practice 641
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Lesson 12.5 Skills Practice page 3
Name Date
17. 6 x 2 1 24x 1 24 5 0
18. 4 x 2 2 9 5 0
19. t 2 1 22t 1 121 5 0
20. 12 w 2 2 48w 1 485 0
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642 Chapter 12 Skills Practice
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Determine the zero(s) of each quadratic function. Check your answer(s).
21. f(x) 5 x 2 2 225
f(x) 5 x 2 2 225 Check:
0 5 x 2 2 225 (215 ) 2 2 225 0 0 (15 ) 2 2 225 0 0
0 5 (x 1 15)(x 2 15) 225 2 225 0 0 225 2 225 0 0
x 1 15 5 0 or x 2 15 5 0 0 5 0 0 5 0
x 5 215 or x 5 15
The zeros are 215 and 15.
22. f(x) 5 x 2 1 x 1 1 __ 4
23. f(x) 5 9 x 2 2 1
Lesson 12.5 Skills Practice page 4
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Chapter 12 Skills Practice 643
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Lesson 12.5 Skills Practice page 5
Name Date
24. f(x) 5 8 x 2 2 48x 1 72
25. f(x) 5 8 x 2 2 50
26. f(x) 5 2 x 2 1 52x 1 338
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644 Chapter 12 Skills Practice
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Chapter 12 Skills Practice 645
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Lesson 12.6 Skills Practice
Name Date
Could It Be Groovy to Be a Square?Approximating and Rewriting Radicals
Vocabulary
Choose the word that best completes each statement.
square root positive (principal) square root radicand
negative square root extract the square root radical expression
1. When solving certain quadratic equations, it is necessary to from both sides of the equation.
2. Every positive number has both a(n) and a(n) .
3. The is the expression enclosed within a radical symbol.
4. A number b is a(n) of a if b 2 5 a.
5. An expression involving a radical symbol is called a(n) .
Problem Set
Rewrite each radical by extracting all perfect squares.
1. √___
25
√___
25 5 65
2. √____
144
3. √____
400 4. √___
12
5. √___
32 6. √___
45
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646 Chapter 12 Skills Practice
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Lesson 12.6 Skills Practice page 2
7. √____
300 8. 5 √___
54
Determine the approximate value of each radical expression to the nearest tenth.
9. √__
7
2. 6 2 5 6.76
2. 7 2 5 7.29
√__
7 < 2.6
10. √___
37
11. √___
96 12. √___
27
13. √____
109 14. √____
405
Solve each quadratic equation. Approximate the roots to the nearest tenth.
15. x 2 5 40
x 2 5 40
√__
x 2 5 6 √___
40
x 5 6 √___
40
6. 3 2 5 39.69
6. 4 2 5 40.96
√___
40 < 66.3
x < 66.3
The roots are approximately 6.3 and 26.3.
16. m 2 5 68
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Chapter 12 Skills Practice 647
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Lesson 12.6 Skills Practice page 3
Name Date
17. t 2 5 15 18. x 2 5 83
19. (x 2 5) 2 5 22 20. (x 1 8) 2 5 29
Solve each quadratic equation. Rewrite the roots in radical form.
21. x 2 5 48
x 2 5 48
√__
x 2 5 6 √___
48
x 5 6 √___
48
x 5 6 √______
16 ? 3
x 5 6 √___
16 ? √__
3
x 5 64 √__
3
The roots are 4 √__
3 and 24 √__
3 .
22. x 2 5 52
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648 Chapter 12 Skills Practice
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Lesson 12.6 Skills Practice page 4
23. x 2 5 27 24. x 2 5 175
25. (12 2 x) 2 5 8 26. (x 1 20) 2 5 80
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Chapter 12 Skills Practice 649
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Another MethodCompleting the Square
Vocabulary
Define the term in your own words.
1. completing the square
Problem Set
Use a geometric figure to complete the square for each expression. Factor the resulting trinomial.
1. x 2 1 2x
x xx2
11
1
x
x
x 2 1 2x 1 1 5 (x 1 1) 2
2. x 2 1 4x
3. x 2 1 12x 4. x 2 1 9x
Lesson 12.7 Skills Practice
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650 Chapter 12 Skills Practice
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Lesson 12.7 Skills Practice page 2
5. x 2 1 11x 6. x 2 1 28x
Determine the unknown value that would make each trinomial a perfect square.
7. x 2 2 10x 1 25 8. x 2 1 14x 1
9. x 2 1 x 1 9 10. x 2 2 x 1 81
11. x 2 1 7x 1 12. x 2 2 15x 1
13. x 2 2 x 1 169 14. x 2 1 x 1 9 __ 4
Determine the roots of each quadratic equation by completing the square. Round your answer to the nearest hundredth. Check your answer.
15. x 2 1 4x 2 6 5 0
x 2 1 4x 2 6 5 0
x 2 1 4x 5 6
x 2 1 4x 1 4 5 6 1 4
(x 1 2) 2 5 10
√_______
(x 1 2) 2 5 6 √___
10
x 1 2 5 6 √___
10
x 5 22 6 √___
10
x < 1.16 or x < 25.16
Check:
(1.16) 2 1 4(1.16) 2 6 0 0
1.3456 1 4.64 2 6 0 0
20.0144 < 0
(25.16) 2 2 4(25.16) 2 6 0 0
26.6256 2 20.64 2 6 0 0
20.0144 < 0
The roots are approximately 1.16 and 25.16.
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Chapter 12 Skills Practice 651
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Lesson 12.7 Skills Practice page 3
Name Date
16. x 2 2 2x 2 4 5 0
17. x 2 1 10x 1 2 5 0
18. x 2 2 12x 1 25 5 0
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652 Chapter 12 Skills Practice
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Lesson 12.7 Skills Practice page 4
19. x 2 1 3x 2 1 5 0
20. x 2 1 x 2 10 5 0