lesson 12.1 skills practice - central cusd 4...chapter 12 skills practice 619 12 lesson 12.1 skills...

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Chapter 12 Skills Practice 617

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Lesson 12.1 Skills Practice

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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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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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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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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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45. Evaluate h(x) at x 5 22. 46. Evaluate h(x) at x 5 21.5.

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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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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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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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21. (x 1 2)( x 2 1 6x 2 1)

22. (x 2 4)( x 2 1 2x 2 3)

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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


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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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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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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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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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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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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


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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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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


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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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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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18. f(x) 5 1 __ 2 x 2 2 3 __

4 x

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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


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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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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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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


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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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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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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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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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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


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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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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

lesson 12.1 skills practice - central cusd 4...chapter 12 skills practice 619 12 lesson 12.1 skills...

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