Cap05

Name:

Chapter 5 Pretest Form A

Date:

For problems 1–3, use the polynomial 5 − 2 x3 + 4 x 2 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

Perform each operation. 4. (5 x 2 − 4 x + 3) + (2 x − 8 x 2 − 9)

4. ____________________________

5. (7 x3 − 6 x + 4) − ( x 2 + 5 x3 − x)

5. ____________________________

6. (4 xy 5 )(−2 x 2 y 4 )

6. ____________________________

7.

( n − 3)( n − 7 )

(

8. −5n3 2n 2 − n + 4

7. ____________________________

)

8. ____________________________

9. (35 x 2 − 15 x + 20) ÷ 5 x

9. ____________________________

Factor completely. 10. 12 x3 − 20 x 2 + 28 x

10. ____________________________

11. x 2 − 8 x + 12

11. ____________________________

12. 49n 2 − 16

12. ____________________________

13. 4 x 2 − 49

13. ____________________________

14. x 2 − 18 x + 81

14. ____________________________

Solve. 15. x 2 − 28 = −3x

15. ____________________________

16. 2 x 2 − 5 x − 33 = 0

16. ____________________________

178

Name:

Chapter 5 Pretest Form B

Date:

For problems 1–3, use the polynomial 7 x − 6 + 9 x 2 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

Perform each operation. 4. (11x 2 − 5 x + 4) + (6 x − 15 x 2 − 12)

4. ____________________________

5. (10 x3 − 7 x 2 − 8) − ( x 2 − 4 x − 7)

5. ____________________________

6. (9 x3 y 4 )(−3xy 2 )

6. ____________________________

7. (x – 5)(3x + 4)

7. ____________________________

8. 8 xy ( x 2 − 3xy − 9 y )

8. ____________________________

9. (6n3 + 10n2 − 2n) ÷ 2n

9. ____________________________

Factor completely. 10. 27 x3 − 18 x 2 + 12 x

10. ____________________________

11. x 2 − 3 x − 28

11. ____________________________

12. 6 x 2 + 17 x + 5

12. ____________________________

13. 25 x 2 − 36

13. ____________________________

14. x 2 − 20 x + 100

14. ____________________________

Solve. 15. x 2 − 13x = −40

15. ____________________________

16. 3x 2 − 19 x + 20 = 0

16. ____________________________

179

Mini-Lecture 5.1 Addition and Subtraction of Polynomials Learning Objectives: 1. 2. 3. 4. 5.

Find the degree of a polynomial. Evaluate polynomial functions. Understand graphs of polynomial functions. Add and subtract polynomial functions. Key vocabulary: terms, degree of a term, polynomial, monomial, binomial, trinomial, leading term, leading coefficient, linear, quadratic, cubic, descending order, polynomial function

Examples: 1. For each polynomial, give the number of terms, the degree of the polynomial, the leading term and the leading coefficient. b) 2 xy 3 − 5 x 2 y 3 + x 2 y 2 a) 2n3 − 4n2 + 6n + 5 2. Write the following polynomials in descending order. b) 6 + 5n3 − 6n + 4n2 a) 7 y − 5 y 2 + 3 3. For the polynomial function P( x) = −2 x3 + 4 x 2 − x + 3, find a) P(4) b) P (−2) 4. Suppose the function f (t ) = $20,700 − $345t gives the amount of money still owed on a loan after t months of payments. How much is still owed on the loan after the following amount of time making payments? a) 10 months b) 4 years 5. Simplify. a) (2 x + 5) + (7 x − 2) c) (−3a 2 − 4) − (a 2 + 8)

b) (3n2 − n + 4) + (10n2 + 5n − 7) d) (7n3 − 5n + 2) − (−4n3 + n2 − 3)

6. The length and width of a rectangle are denoted by 6n − 1 and 2n 2 + 5n + 1 respectively. Find an expression for the perimeter of the rectangle. Teaching Notes: • Students often confuse finding the degree of a polynomial with finding the degree of a term. When finding the degree of a polynomial, it is tempting for students to find the sum of the degrees of each its terms instead of taking the highest degree of its terms. Answers: 1a) 4, 3, 2n3, 2; 1b) 3, 5, –5x2y3, –5; 2a) −5y 2 + 7y + 3 ; 2b) 5n3 + 4n 2 − 6n + 6 ; 3a) –65; 3b) 37; 4a) $17,250; 4b) $4140; 5a) 9x + 3 ; 5b) 13n 2 + 4n − 3 ; 5c) −4a 2 − 12 ; 5d) 11n3 − n 2 − 5n + 5 ; 6) 4n 2 + 22n 180

Mini-Lecture 5.2 Multiplication of Polynomials Learning Objectives: 1. 2. 3. 4. 5. 6. 7.

Multiply a monomial by a polynomial. Multiply a binomial by a binomial. Multiply a polynomial by a polynomial. Find the square of a binomial. Find the product of the sum and difference of the same two terms. Find the product of polynomial functions. Key vocabulary: factors, distributive property, FOIL method, difference of two squares

Examples: 1. Multiply. a) 3ab3 −4a 4b3

(

)(

)

(

b) 5n3 2n 2 − n + 4

2. Multiply. a) ( x + 6)( x + 8)

b) (5n − 1)(n − 4)

3. Multiply. a) ( x − 2 ) x 2 + 3 x + 1

(

)

)

b)

(

c) 3 xy 2 −3 x 2 y 2 + xy − 5 y

)

c) (2 y + 5)(3 y − 1)

( 2 x − 1) ( x3 + 3x 2 − 5 x + 6 )

4. Multiply using either the formula for the square of a binomial or for the product of the sum and difference of the same two terms.

(x

)

2

( x + 4 )2

b)

( 3 y − 5 )2

c)

d) [( x − 2) + y ]2

e)

( n + 6 )( n − 6 )

f) ( 5m − 2n )( 5m + 2n )

a)

2

+7

g) [3a − (b + 4)][3a + (b + 4)] 5. Let f ( x) = 2 x + 1 and g ( x) = x − 2. Find a) f (5) ⋅ g (5) b) ( f ⋅ g )( x)

c) ( f ⋅ g )(5)

Teaching Notes:

• Mention to students the difference between problems like ( x + 6)( x + 8) that require the FOIL method and problems like x + 6( x + 8) that do not require the FOIL method.

Answers: 1a) −12a 5b6 ; 1b) 10n5 − 5n4 + 20n3 ; 1c) −9x3 y 4 + 3x 2 y 3 − 15xy3 ; 2a) x 2 + 14x + 48 ; 2b) 5n 2 − 21n + 4 ; 2c) 6y 2 + 13y − 5; 3a) x3 + x 2 − 5x − 2 ; 3b) 2x 4 + 5x3 − 13x 2 + 17x − 6 ; 4a) x 2 + 8x + 16 ; 4b) 9y 2 − 30y + 25 ; 4c) x 4 + 14x 2 + 49; 4d) x 2 + 2xy − 4x + y 2 − 4y + 4 ; 4e) n 2 − 36; 4f) 25m 2 − 4n 2 ; 4g) 9a 2 − b 2 − 8b − 16; 5a) 33 5b) 2x 2 − 3x − 2; 5c) 33

181

Mini-Lecture 5.3 Division of Polynomials and Synthetic Division Learning Objectives:

1. 2. 3. 4. 5.

Divide a polynomial by a monomial. Divide a polynomial by a binomial. Divide polynomials using synthetic division. Use the Remainder Theorem. Key vocabulary: dividend, divisor, synthetic division, Remainder Theorem

Examples:

1. Divide. y9 a) y5 d)

12 x + 20 4

−9 x 4 y 5 c) 6 xy 3

a 7b 5 b) 4 2 ab

e)

50 x5 + 30 x 4 − 25 x3 − 5 x 2 5x2

2. Divide using long division. 6 x2 + 5x − 6 3x 2 − 8 x + 3 b) a) 2x + 3 3x − 2

f)

7c3 − 21c 2d + 14cd 2 − 35d 3 14cd

6 y 3 + 21y 2 − 20 c) 2y + 3

3. Divide using synthetic division. x3 − 2 x 2 − 4 x + 5 3x3 + 9 x 2 − 11x + 4 b) a) x−3 x+4

c)

3 x 4 − 8 x3 + 9 x + 5 x−2

4. Determine the remainder for the following divisions using the Remainder Theorem. b) ( x5 + 5 x 4 + 6 x3 − x 2 + 4 x + 29) ÷ ( x − 1) a) (2 x 4 + 5 x3 − 2 x − 8) ÷ ( x + 3)

Teaching Notes:

• Remind students that synthetic division does not work when dividing a polynomial by a binomial of the form x 2 − a.

−3x 3 y 2 ; 1d) 3x + 5; 1e) 10x 3 + 6x 2 − 5x − 1; 2 c 2 3c 5d 2 1 7 − +d − ; 2a) 3x − 2; 2b) x − 2 − ; 1f) ; 2c) 3y 2 + 6y − 9 + 2d 2 2c 3x − 2 2y + 3 2 7 ; 4a) 25; 4b) –3 3a) x 2 + x − 1 + ; 3b) 3x 2 − 3x + 1; 3c) 3x 3 − 2x 2 − 4x + 1 + x−3 x−2 Answers: 1a) y 4 ; 1b) a 3b3 ; 1c)

182

Mini-Lecture 5.4 Factoring a Monomial from a Polynomial and Factoring by Grouping Learning Objectives:

1. 2. 3. 4. 5.

Find the greatest common factor. Factor a monomial from a polynomial. Factor a common binomial factor. Factor by grouping. Key vocabulary: greatest common factor (GCF), factor by grouping

Examples:

1. State the greatest common factor of the following terms. b) 10(n + 1)3 , 40(n + 1), 5(n + 1)4 a) 12a3b4c 2 , 18a 2bc3 , 24a3b5c 2 2. Factor out the greatest common factor. a) 24 x − 8

b) 16n 4 − 12n3 + 10n 2

c) −3x3 − 21x 2 − 15 x

d) 12 x3 y − 8 x 2 y 2 + 8 xy 3

e) 3n(a + 3) + 7(a + 3)

f) x(5 y − 1) − 7(5 y − 1)

3. Factor by grouping. a) 10mp + 5np + 4mq + 2nq

b) p 2q 2 − 2q 2 + 5 p 2 − 10 d) a 2 + 7b + ab + 7a

c) 28 xy + 21 y − 36 x − 27

Teaching Notes:

• When factoring by grouping, it may be necessary to helpful to rearrange the terms if there is no common factor in the first pair of terms.

Answers: 1a) 6a 2bc 2 ; 1b) 5(n + 1); 2a) 8(3x − 1); 2b) 2n 2 (8n2 − 6n + 5); 2c) −3x(x 2 + 7x + 5); 2d) 4xy(3x 2 − 2xy + y 2 ); 2e) (a + 3)(3n + 7); 2f) (5y − 1)(x − 7); 3a) (2m + n)(5p + 2q); 3b) (p 2 − 2)(q 2 + 5); 3c) (4x + 3)(7y − 9); 3d) (a + b)(a + 7)

183

Mini-Lecture 5.5 Factoring Trinomials Learning Objectives: 1. Factor trinomials of the form x 2 + bx + c. 2. Factor out a common factor. 3. Factor out trinomials of the form ax 2 + bx + c, a ≠ 1, using trial and error.

4. Factor trinomials of the form ax 2 + bx + c, a ≠ 1, using grouping. 5. Factor trinomials using substitution. 6. Key vocabulary: trinomial, trial and error method, grouping method, prime polynomial, factoring using substitution Examples:

1. Factor each trinomial completely. If the polynomial is prime, so state. b) x 2 + 5 x + 6 c) n 2 − 12n + 32 a) y 2 + 2 y − 24 d) x 2 − 7 x − 44

e) 3 y 2 − 27 y + 54

g) x 2 + 7 xy + 10 y 2

h) n 2 + 9n − 18

f) 2 x 2 − 2 x − 24

2. Factor using the trial and error method. b) 5n2 − 7n + 2 a) 3x 2 + 17 x + 10

c) 4 x 2 + 13x + 3

3. Factor using the grouping method. a) 7 x 2 − 25 x − 32 b) 4 x 2 − 22 x + 10

c) 14 x 2 + 7 x − 21

4. Factor each trinomial completely. a) 2 x5 + 4 x3 − 16 x b) − x 2 − 14 x − 40

c) x 4 − 2 x 2 − 3

d) −8 x3 + 22 x 2 − 12 x

e) ( x + 3)2 − 3( x + 3) − 10

Teaching Notes:

• It is often helpful to factor a “–1” from the entire trinomial when using the trial and error method to factor a trinomial with a negative leading coefficient. • Forgetting to look for a common factor first is one of the most common mistakes that students make when factoring. Answers: 1a) (y + 6)(y – 4); 1b) (x + 3)(x + 2); 1c) (n – 8)(n – 4); 1d) (x + 4)(x – 11); 1e) 3(y – 6)(y – 3); 1f) 2(x + 3)(x – 4); 1g) (x + 5y)(x + 2y); 1h) prime; 2a) (3x + 2)(x + 5); 2b) (5n – 2)(n – 1); 2c) (4x + 1)(x + 3); 3a) (x + 1)(7x – 32); 3b) 2(2x – 1)(x – 5); 3c) 7(2x + 3)(x – 1); 4a) 2x(x 2 + 4)(x 2 − 2); 4b) −(x + 10)(x + 4); 4c) (x 2 + 1)(x 2 − 3); 4d) –2x(x – 2)(4x – 3); 4e) (x + 5)(x – 2) 184

Mini-Lecture 5.6 Special Factoring Formulas Learning Objectives:

1. 2. 3. 4.

Factor the difference of two squares. Factor perfect square trinomials. Factor the sum and difference of two cubes. Key vocabulary: difference of two squares, perfect square trinomial, sum of two cubes, difference of two cubes

Examples:

1. Factor using the difference of two squares formula. b) n2 − 36 a) x 2 − 9 d) 6c 2 − 6

e) 20n − 45n3

g) (b + 4)2 − 9

h) 25 − (2 x + 7)2

2. Factor using the perfect square trinomial formula. b) n 2 − 20n + 100 a) x 2 + 14 x + 49 d) 9a 4 + 24a 2 + 16

e) (m − n)2 − 6(m − n) + 9

3. Factor using the sum or difference of two cubes formula. b) x3 − 8 a) x3 + 8 d) 2 y 3 − 250

e) 2a3b + 54b4

c) 16 x 2 − 49 y 2 f) n 4 − 16

c) 4 x 2 + 12 xy + 9 y 2 f) x 2 + 8 x + 16 − 81y 2

c) 8n3 + 27 f) (n − 2)3 − 8

Teaching Notes:

• For students who think that x 2 + 4 can be factored as ( x + 2)2 , have them FOIL out ( x + 2)2

to show they obtain x 2 + 4 x + 4 instead of x 2 + 4.

Answers: 1a) (x + 3)(x – 3); 1b) (n + 6)(n – 6); 1c) (4x + 7y)(4x – 7y); 1d) 6(c + 1)(c – 1); 1e) 5n(2 + 3n)(2 – 3n); 1f) (n2 + 4)(n + 2)(n − 2); 1g) (b + 1)(b + 7); 1h) –4(x + 6)(x + 1); 2a) (x + 7)2 ; 2b) (n − 10)2 ; 2c) (2x + 3y)2 ; 2d) (3a 2 + 4)2 ; 2e) (m − n − 3)2 ; 2f) (x + 4 – 9y)(x + 4 + 9y); 3a) (x + 2)(x 2 − 2x + 4); 3b) (x − 2)(x 2 + 2x + 4); 3c) (2n + 3)(4n 2 − 6n + 9); 3d) 2(y − 5)(y 2 + 5y + 25); 3e) 2b(a + 3b)(a 2 − 3ab + 9b2 ); 3f) (n − 4)(n 2 − 2n + 4)

185

Mini-Lecture 5.7 A General Review of Factoring Learning Objectives:

1. Factor polynomials using a combination of techniques. Examples:

1. Factor each polynomial completely. b) x 2 + xy − 2 x − 2 y a) 4 x3 − 25 x

c) x 6 − 8

d) 2 x5 + 6 x3 − 8 x

e) m2n − 4n + 5m2 − 20

f) 27n3 + a 6b3

g) 5 x3 + 60 x 2 + 180 x

h) 4 x 2 + 8 x + 4 − 36 y 2

i) 3x 2 y − xy 2 − 10 y 3

Teaching Notes:

• Once again, failing to factor the greatest common factor in the first step could be crucial. Difficult factoring problems will become much easier once the GCF has been factored out. Some polynomials, especially binomials, cannot be completely factored until the GCF has been dealt with.

Answers: 1a) x(2x + 5)(2x − 5); 1b) (x + y)(x − 2); 1c) (x 2 − 2)(x4 + 2x 2 + 4); 1d) 2x(x 2 + 4)(x + 1)(x − 1); 1e) (m + 2)(m − 2)(n + 5); 1f) (3n + a 2b)(9n2 − 3na 2b + a 4 b2 ); 1g) 5x(x + 6)2 ; 1h) 4(x + 1 − 3y)(x + 1 + 3y); 1i) y(3x + 5y)(x − 2y)

186

Mini-Lecture 5.8 Polynomial Equations Learning Objectives:

1. 2. 3. 4. 5.

Use the zero-factor property to solve equations. Use factoring to solve equations. Use factoring to solve applications. Use factoring to find the x-intercepts of a quadratic function. Key vocabulary: polynomial equation, degree of a polynomial equation, quadratic equation, standard from, zero-factor property, cubic equation, hypotenuse, Pythagorean Theorem, x-intercept

Examples:

1. Solve. a) ( x + 4)(2 x − 3) = 0

b) 6 x( x − 1)( x + 5) = 0

c) x 2 − 4 x − 12 = 0

d) 3x 2 + 7 x + 2 = 0

e) 3n 2 − 21n + 30 = 0

f) 4n3 = 36n

g) x 2 + 13x = 6 x − 10

h) y ( y − 2) = 24

i) ( x − 3)( x − 5) = 8

2. Solve the following applied problems. a) The sum of a number and its square is 20. Find the number. b) A football is kicked in the air such that its height above the ground, in feet, after t seconds is given by the function h(t ) = −16t 2 + 64t. Determine how long the football is in the air. c) The base of a triangle is 2 centimeters more than twice its height. The area of the triangle is 30 square centimeters. Find the base and height of the triangle. d) Let h(t ) = t 2 − 7t + 19. Find all values of t for which h(t ) = 9. e) Find the lengths of the sides of a right triangle if the legs are x and x – 2 and the hypotenuse is x + 2. 3. Find the x-intercepts of the following functions. b) y = 6 x 2 − 5 x + 1 a) y = x 2 + 8 x + 15

c) y = 5 x3 − 6 x 2 + x

Teaching Notes:

• With an expression, common factors can be factored from the expression but they are not eliminated from the problem. When solving equations, common numerical factors can be eliminated by dividing both sides of the equation by that particular number.

187

Mini-Lecture 5.7 A General Review of Factoring 3 2

1 3

Answers: 1a) −4, ; 1b) 0, 1, –5; 1c) –2, 6; 1d) −2, − ; 1e) 2, 5; 1f) –3, 0, 3; 1g) –5, –2; 1h) –4, 6; 1i) 1, 7; 2a) –5 or 4; 2b) 4 seconds; 2c) height = 5 cm, base = 12 cm; 2d) 2, 5; 2e) 6, 8, 10; 3a) –5, –3; 3b)

1 1 1 , ; 3c) 0, ,1 5 3 2

188

Mini-Lecture 5.8 Polynomial Equations Perform the operation. 1. 13a − 8a

1. ____________________________

2. 5 x3 y 2 z − 2 x3 y 2 z

2. ____________________________

3. 4abc − 10abc

3. ____________________________

4. 3x 2 y − 5 xy 2 − 7 x 2 y

4. ____________________________

5. 3ab3 − 4a 2b 2 − 2ab3 + 2a 2b 2

5. ____________________________

6.

( 3x y ) 2

2

+ 2x4 y 2 − 5x4 y 2

6. ____________________________

7. (3x + 7) + (4 x − 3)

7. ____________________________

8.

( −2 a

2

) (

9.

(−x

2

+ 2 x + 3 − 4 x 2 − 3x − 1

10.

(a

− 3a + 5 − 3a 2 + a + 2

11.

(7x

3

+ 2 x 2 + 1 + −8 x 3 − x + 1

12.

(6x

3

+ 2 x + 4 − 2 x3 + 3 x 2 + 3

− 5a + 3 + 3a 2 + 2a − 4

) (

8. ____________________________

)

) (

2

)

9. ____________________________

)

10. ____________________________

) (

)

11. ____________________________

) (

)

12. ____________________________

13. 3 ( x + 2 ) + 2 ( x − 3)

13. ____________________________

14. −2 ( x − 4 ) + 3 ( x + 1)

14. ____________________________

15. −8( x − y ) + 11( x − y )

15. ____________________________

16. 4 ( y + 5) − 3 ( y + 2 )

16. ____________________________

(

) (

17. −5 2 x3 + x 2 + x + 2 3x3 − x 2 − x

(

) (

)

17. ____________________________

)

18. −3 x 2 + y 2 + 1 − 4 2 x 2 − y 2 − 1 19.

( −2 x

2

)

19. ____________________________

20.

(7x

− 4 x − x 2 + 2 + ( 5 + 3x )

20. ____________________________

3

) (

18. ____________________________

− x + 7 − 3x 2 + 4 x − 3

) (

)

189

Name:

Additional Exercises 5.1

Date:

Perform the operation. 1.

( 2 x ) ( −3xy )

2.

( −3x y )(3xy )

2. ____________________________

3. (−2a 2b3 )(5ab2 )

3. ____________________________

4.

2

2

(x

2 2

1. ____________________________ 2

)(

y z xy 3 z 2

)

4. ____________________________

5. 3 ( x + 2 )

(

5. ____________________________

)

6. −5 x 2 + 2 x +1

6. ____________________________

7. y 2 ( −3 y − 4 )

7. ____________________________

(

)

8. −2 x 3x 2 + x − 1

8. ____________________________

9. 2n 2 (3n2 − 4n)

9. ____________________________

(

10. 5 xy 2 2 x 2 y − 3xy 3

)

10. ____________________________

11.

( x + 2 )( x + 3)

11. ____________________________

12.

( z − 7 )( z − 4 )

12. ____________________________

13.

( 2a + 1)( a − 3)

13. ____________________________

14.

( 2 x − 4 )( 3x + 2 )

14. ____________________________

15.

( p + 3)( 2 p − 4 )

15. ____________________________

16.

( x − 3 )2

16. ____________________________

17.

( y + 6 )2

17. ____________________________

18.

( x − y ) ( x 2 + xy + y 2 )

18. ____________________________

19.

( x + 3 y ) ( x2 − 3xy + 9 y 2 )

19. ____________________________

20.

( 2a − b ) ( 4a 2 + 2ab + b2 )

20. ____________________________

190

Name:

Additional Exercises 5.2

Date:

Perform the operation. 1.

(6x

3

)

2.

( 4x

2 3

3.

( 24 x y

4.

(x

5.

(8a

6.

+ 8x2 − 4 x ÷ ( 2x )

1. ____________________________

)

y + x3 y 2 ÷ ( xy )

6 7

2. ____________________________

) (

− 12 x5 y12 + 36 xy ÷ 4 x 2 y 3

)

3. ____________________________

+ 10 x + 21 ÷ ( x + 3)

)

4. ____________________________

2

+ 2a − 3 ÷ ( 2a − 1)

)

5. ____________________________

(6x

2

− 11x + 2 ÷ ( 3 x − 1)

)

6. ____________________________

7.

(6x

3

+ 11x 2 − x − 2 ÷ ( 3 x − 2 )

8.

(6x

3

− x 2 − 6 x − 9 ÷ ( 2 x − 3)

9.

( 2 x + 1 + x ) ÷ ( x + 1)

2

)

7. ____________________________

)

8. ____________________________

2

9. ____________________________

10.

( 6 y + 10 y

11.

(6x

2

− 18 x + 12 ÷ ( x − 1)

12.

(9x

3

− 9 x 2 + 8 x − 4 ÷ ( 3x − 2 )

13.

(x

− 5 x + 14 ÷ ( x + 2 )

14.

( 3x

15.

( 2x

3

− 5x − 6 ÷ ( x − 2)

16.

(6x

3

+ 5 x 2 + 4 ÷ ( x + 1)

17.

(6x

3

+ x 2 + 2 x + 1 ÷ ( 3 x −1)

18.

( 3x

2

+ 13xy − 10 y 2 ÷ ( 3 x − 2 y )

19.

(8x

3

− 27 y 6 ÷ 2 x − 3 y 2

20.

(x

)

10. ____________________________

)

11. ____________________________

− 4 x 2 + 5 x + 3 ÷ ( x + 1)

− 4 ÷ (5 y − 2)

)

12. ____________________________

)

2

3

2

3

13. ____________________________

)

− 10 x 2 + 5 x − 6 ÷ ( x − 3)

14. ____________________________

)

15. ____________________________

)

16. ____________________________

)

17. ____________________________

)

) (

18. ____________________________

)

19. ____________________________

)

20. ____________________________

191

Name:

Additional Exercises 5.3

Date:

1. Find the greatest common factor of 14x, 7x 2 , and 21x 4 .

1. ____________________________

2. Find the greatest common factor of 2x 2 y 4 , 6x 2 y 5 , and 2x 2 y 3 .

2. ____________________________

3. Find the GCF of the following terms. 19(t + 8), 7(t + 8), 20(t + 8)2

3. ____________________________

4. Find the GCF of the following terms. 13(t + 3)2 , 16(t + 3), 15(t + 3)2

4. ____________________________

5. Factory completely: 6 x3 + 3x

5. ____________________________

6. Factor completely: 18 x 4 y 3 − 30 x3 y 2 + 24 x5 y 4

6. ____________________________

7. Factor: 12 x5 − 15 x 4 + 9 x3 + 15 x 2

7. ____________________________

8. Factor completely: 4 y 2 + 8 y − 2 xy

8. ____________________________

9. Factor by grouping: 4 x 2 + 4 x + 3x + 3

9. ____________________________

10. Factor by grouping: 3ab + 9a − 2b − 6

10. ____________________________

11. Find the greatest common factor of 16x, 8x 2 , and 24x 4 .

11. ____________________________

12. Find the greatest common factor of x 2 y 9 , x8 y8 , and x 7 y 3 .

12. ____________________________

13. Find the GCF of the following terms. 2(t + 7)4 , 17(t + 7)4 , 19(t + 7)3

13. ____________________________

14. Find the GCF of the following terms. 11(t − 4), 20(t − 4)4 , 7(t − 4)3

14. ____________________________

15. Factor completely: 16 x 2 − 40 x5

15. ____________________________

16. Factor completely: 35 x 4 y 4 + 42 x3 y 3 − 49 x5 y 7

16. ____________________________

17. Factor: 12 x6 − 15 x5 + 9 x 4 + 15 x3

17. ____________________________

18. Factor completely: 5 x3 + 15 x 2 + 15 x

18. ____________________________

19. Factor by grouping: 4 x 2 + 24 x + 3x + 18

19. ____________________________

20. Factor by grouping: x 2 − 7 x + 3x − 21

20. ____________________________

192

Name:

Additional Exercises 5.4

Date:

Factor completely. 1. 3x 2 + 6 x + x + 2

1. ____________________________

2. 3x 2 + 18 x + 2 x + 12

2. ____________________________

3. x 2 − 5 x + 4

3. ____________________________

4. x 2 − 11x + 18

4. ____________________________

5. x 2 + x − 30

5. ____________________________

6. x 2 + 7 x + 12

6. ____________________________

7. 4 x 2 − 16 xy + 15 y 2

7. ____________________________

8. 15 x 2 − 26 xy + 8 y 2

8. ____________________________

9. 15 x 2 + 2 + 11x

9. ____________________________

10. 12 x 2 − 25 x + 12

10. ____________________________

11. 24 p 2 − 2 p − 15

11. ____________________________

12. 3x 2 + x − 4

12. ____________________________

13. 21a 2 − 4a − 12

13. ____________________________

14. 15c 2 − 2c − 1

14. ____________________________

15. 5 x3 − 5 x 2 − 30 x

15. ____________________________

16. 9 x3 − 81x

16. ____________________________

17. 5 x 2 − 14 x − 24

17. ____________________________

18. −18 x 2 − 3x + 45

18. ____________________________

19. x 4 − x 2 − 30

19. ____________________________

20. ( x − 5)2 + 7( x − 5) + 6

20. ____________________________

193

Name:

Additional Exercises 5.5

Date:

Factor completely. 1. 36 j 2 − k 2

1. ____________________________

2. 36 x 6 − 25 y 2

2. ____________________________

3. x 2 − 49

3. ____________________________

4. x 2 − 36

4. ____________________________

5. 4 p 2 + 20 p + 25

5. ____________________________

6. 16 x 2 y 2 + 24 xy + 9

6. ____________________________

7. x 2 − 6 x + 9

7. ____________________________

8. x 2 − 24 x + 144

8. ____________________________

9. 8c3 − 125

9. ____________________________

10. x3 − 64

10. ____________________________

11. x 21 + y 24

11. ____________________________

12. x 27 + y 30

12. ____________________________

13. 4 x5 y 3 − 256 x5

13. ____________________________

14. 81a 4 − 16b 2

14. ____________________________

15. 9r 2 − 4s 2

15. ____________________________

16. 16 x 6 − 9 y 2

16. ____________________________

17. x 2 − 100

17. ____________________________

18. 5 x3 − 125 x

18. ____________________________

19. 27 x3 + 125

19. ____________________________

20. x3 + 512

20. ____________________________

194

Name:

Additional Exercises 5.6

Date:

Factor completely. 1. ( x − 4)2 + ( x − 4) − 20

1. ____________________________

2. x 4 − x 2 − 12

2. ____________________________

3. 8 x3 y − 30 x 2 y 2 − 8 xy 3

3. ____________________________

4. 12 x3 y − 57 x 2 y 2 − 15 xy 3

4. ____________________________

5. 3x3 − 12 xy 2

5. ____________________________

6. 25 x 2 − 30 xy + 8 y 2

6. ____________________________

7. 25 x 2 − 20 xy + 4 y 2

7. ____________________________

8. 20 x 2 + 6 + 23x

8. ____________________________

9. 15 x 2 + 20 + 37 x

9. ____________________________

10. 18m2 − 9m − 20

10. ____________________________

11. 6 x 2 + 7 x − 3

11. ____________________________

12. 2t 2 + 15t + 18

12. ____________________________

13. 12 p 2 − 8 p − 15

13. ____________________________

14. 3x3 − 3x 2 − 6 x

14. ____________________________

15. 6 x3 − 18 x 2 − 108 x

15. ____________________________

16. 60 x 2 + 39 x − 45

16. ____________________________

17. 45 x 2 − 42 x − 24

17. ____________________________

18. x 4 − 17 x 2 + 16

18. ____________________________

19. ( x + 4)2 − 4(2 x + 3) − 5

19. ____________________________

20. ( x + 2)2 − 2( x + 2) − 3

20. ____________________________

195

Name:

Additional Exercises 5.7

Date:

Solve. 1. x 2 + 6 x = 0

1. ____________________________

2. x 2 − 2 x + 1 = 0

2. ____________________________

3. 144x = x 2

3. ____________________________

4. 15 x 2 = 20 x

4. ____________________________

5. 3x 2 + 11x = 20

5. ____________________________

6. x3 + 2 x 2 − 8 x = 0

6. ____________________________

7. 2 x 2 + x − 3 = 0

7. ____________________________

8. 3x 2 + 14 x = 5

8. ____________________________

9. ( x − 2)2 − 3 = −3

9. ____________________________

10. 2(4w + 3) 2 + (4w + 3) − 6 = 0

10. ____________________________

11. If each side of a square is increased by 7 meters, the area becomes 121 square meters. Find the length of a side of the original square.

11. ____________________________

12. The area of a triangle is 45 square meters. If the base of the triangle is 3 greater than 2 times the height, find the base and height of the triangle.

12. ____________________________

13. The volume of a box is 1274 cubic feet. The width of the box is 7 feet and its height is 1 foot more than its length. Find the height of the box.

13. ____________________________

14. The product of two consecutive positive integers is 527 more than the next integer. What is the largest of the three integers?

14. ____________________________

15. Find the x intercepts of y = x 2 + 8 x + 15 .

15. ____________________________

16. Find the x-intercepts of y = 16 x 2 − 32 x + 15 .

16. ____________________________

17. Find the x-intercepts of y = x 2 + 3 .

17. ____________________________

18. Find the x-intercepts of the graph y = x3 + 3x 2 − 4 x − 12 .

18. ____________________________

19. Solve: x 2 + 2 x = 0

19. ____________________________

20. Solve: x 2 − 11x + 24 = 0

20. ____________________________

196

Chapter 5 Test Form A (cont.)

Name:

For questions 1–4, refer to the polynomial −3x 2 + 2 x + 5 x3 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

4. What is the leading coefficient of the polynomial?

4. ____________________________

For questions 5–10, perform the operation. 5.

(8x

6.

(1.2 x

4

) (

− 3x 2 + 5 − 2 x3 − 5 x 2 + 5

2

)

) (

y + 2.1y 2 + 0.3 x − 0.2 x 2 y − 0.7 x + 0.8 y 2

(

7. 2 xy 3 3 x 4 − 7 x 2 y + 5 xy 3 − 2

5. ____________________________

)

6. ____________________________

)

7. ____________________________

8. (3 y − 4)( y + 2)

9.

(5x

10.

(6x

3

2

8. ____________________________

− 8 x + 1 ( −4 x − 3)

)

9. ____________________________

)

10. ____________________________

− 2 x + 1 ÷ ( 3x − 1)

If f ( x) = 2 x + 5 and g ( x) = x − 7 , find 11.

( f ⋅ g ) ( x)

11. ____________________________

12.

( f ⋅ g )( −1)

12. ____________________________

13. Use synthetic division to obtain the quotient

( 3x

3

13. ____________________________

)

− 2 x + 6 x − 3 ÷ ( x − 1) . 2

14. Use the remainder theorem to find the remainder when 4 x 2 − 3x + 2 is divided by x + 1.

197

14. ____________________________

Name:

Chapter 5 Test Form A

Date:

For problems 15–20, factor completely. 15. 4 x3 + 4 x 2 + 8 x

15. ____________________________

16. 16 x3 y 5 + 10 x 2 y 4 − 6 x5 y 7

16. ____________________________

17. x 2 − 3x − 18

17. ____________________________

18. 2 x 2 + 9 x − 35

18. ____________________________

19. 2 y ( x + 7) − 5( x + 7)

19. ____________________________

20. m3 + 125

20. ____________________________

21. Find an equation whose graph has x-intercepts of –3 and 5.

21. ____________________________

For questions 22 and 23, solve. 22. 2 x 2 + 3x = 2

22. ____________________________

23. 6 x3 + 12 x = 17 x 2

23. ____________________________

24. The area of a triangle is 104 square meters. If the base of the triangle is 10 greater than 2 times the height, find the base and height of the triangle.

24. ____________________________

nt

⎛ r⎞ 25. The compound interest formula is A = P ⎜ 1 + ⎟ , where A 25. ____________________________ ⎝ n⎠ is the amount, P is the principal invested, r is the annual rate of interest, n is the number of times the interest is compounded annually, and t is the time in years. Find the value of A if P = $2,000, n = 4, r = 8%, and t = 2 years.

198

Chapter 5 Test Form B (cont.)

Name:

For questions 1–4, refer to the polynomial 7 − 4 x5 + 3x . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

4. What is the leading coefficient of the polynomial?

4. ____________________________

For questions 5–10, perform the operation. 5.

(x

6.

( 7 xy

5

) (

− 3 x3 + x − 2 x5 + x 2 − 2 x

2

) (

)

+ 2 x 2 + y − 2 x 2 + 3xy 2 − 5 y

(

7. − x 2 y 3 4 x − 7 x 2 y + 8 y 4

5. ____________________________

)

6. ____________________________

)

7. ____________________________

8. ( x + 8)(2 x − 3)

9.

10.

8. ____________________________

( x + 2 ) ( −3x 2 + 7 x − 1)

(8x

2

9. ____________________________

)

− 4 x − 3 ÷ ( 2 x + 1)

10. ____________________________

If f(x) = x + 4 and g(x) = 8x – 3, find 11.

( f ⋅ g ) ( x)

11. ____________________________

12.

( f ⋅ g )( 2 )

12. ____________________________

13. Use synthetic division to obtain the quotient

( 24 x

2

13. ____________________________

)

− 6 x − 18 ÷ ( x − 1) .

14. Use the remainder theorem to find the remainder when 12 x3 − 7 x 2 − 4 x + 1 is divided by x + 2.

199

14. ____________________________

Name:

Chapter 5 Test Form B

Date:

For problems 15–20, factor completely. 15. x 2 − 3x + 2

15. ____________________________

16. 10 x 2 − 33xy + 20 y 2

16. ____________________________

17. 6 x 2 + 15 x − 9

17. ____________________________

18. x 4 − 6 x 2 − 40

18. ____________________________

19. 64 x 4 − 25 y 2

19. ____________________________

20. x3 − 64

20. ____________________________

21. Find an equation whose graph has x-intercepts of −

1 2 and . 3 2

21. ____________________________

For questions 22 and 23, solve. 22. 9x = x 2

22. ____________________________

23. 4(a 2 − 3) = 6a + 4(a + 3)

23. ____________________________

24. The product of two consecutive positive integers is 359 more than the next integer. What is the largest of the three integers?

24. ____________________________

25. Find the area of the shaded region in the figure.

25. ____________________________

x y

200

Chapter 5 Test Form C (cont.)

Name:

For questions 1–4, refer to the polynomial −8 x5 + 2 x9 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

4. What is the leading coefficient of the polynomial?

4. ____________________________

For questions 5–10, perform the operation. 5. (4a 2 − 7a + 2) − (8 − a − 3a 2 )

6.

(8 x

2

) (

+ 2 x − 5 − 3x2 − 4 x + 2

(

7. −2 x3 y 5 xy − 3x 4 + 2 y 2

5. ____________________________

)

6. ____________________________

)

7. ____________________________

8.

( 4n − 3)( 5n − 8)

8. ____________________________

9.

( 2 x − 1) ( −7 x3 + x − 6 )

9. ____________________________

10.

(6x

)

y + 4 x 2 y − 2 xy ÷ 2 xy

2 2

10. ____________________________

If f(x) = 5x – 3 and g(x) = x – 2, find 11.

( f ⋅ g ) ( x)

11. ____________________________

12.

( f ⋅ g )( −3)

12. ____________________________

13. Use synthetic division to obtain the quotient

(7x

2

13. ____________________________

)

− 41x − 6 ÷ ( x − 6 ) .

14. Use the remainder theorem to find the remainder when 9 x3 − 6 x 2 − 5 x + 3 is divided by x – 4.

201

14. ____________________________

Name:

Chapter 5 Test Form C

Date:

For problems 15–20, factor completely. 15. 12r 2 − 4r − 5

15. ____________________________

16. ax + ay + cx + cy

16. ____________________________

17. 9 x3 − 45 x 2 − 126 x

17. ____________________________

18. x12 − y15

18. ____________________________

19. 25 x 2 y 2 − 60 xy + 36

19. ____________________________

20. 81s 4 − 625t 4

20. ____________________________

21. Find an equation whose graph has x-intercepts of

4 1 and − . 5 5

21. ____________________________

For questions 22 and 23, solve. 22. 16 x 2 + 24 x + 9 = 0

22. ____________________________

23. 9(8w + 2)2 + 3(8w + 2) − 20 = 0

23. ____________________________

24. The area of a rectangle is 18 x 2 − 6 x − 4 . If the width of the rectangle is 3 x − 2 , find the length.

24. ____________________________

25. A ball is thrown upward with a speed of 32 feet per second from the edge of a building 48 feet tall. The ball’s height, h, above the ground at any time, t, is given by the equation h = −16t 2 + 32t + 48 . Find the time it takes the ball to hit the ground.

25. ____________________________

202

Chapter 5 Test Form D (cont.)

Name:

For questions 1–4, refer to the polynomial −9 − 2x3 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

4. What is the leading coefficient of the polynomial?

4. ____________________________

For questions 5–10, perform the operation. 5.

(5x

6.

( −4rt

4

) (

)

5. ____________________________

) (

)

6. ____________________________

+ 7 x 2 − 3 + 4 x3 − 6 x 2 + x

3

+ 2r 2t 2 − −3rt 3 + 2r 2t 2

7. 3( x 2 + 4 x) + 2( x 2 − 4)

7. ____________________________

8. 3(2 x − 3)( x + 1)

8. ____________________________

9. (4 x + 3)( x 2 + 2 x + 3)

9. ____________________________

10.

( 42 x y

3 2

)

+ 36 x 2 y − 12 xy + 1 ÷ 6 xy

10. ____________________________

If f ( x) = 4 x − 5 and g ( x) = 3x + 4 , find 11.

( f ⋅ g ) ( x)

11. ____________________________

12.

( f ⋅ g )( 2 )

12. ____________________________

13. Use synthetic division to obtain the quotient

(

13. ____________________________

)

2 x3 − 5 x 2 + 7 x − 4 ÷ ( x − 1) .

14. Use the remainder theorem to find the remainder when 4 x3 + 8 x 2 − 2 x + 3 is divided by x + 3.

203

14. ____________________________

Name:

Chapter 5 Test Form D

Date:

For problems 15–20, factor completely. 15. 10 x 2 + 20 − 33x

15. ____________________________

16. x 4 − x 2 − 2

16. ____________________________

17. x 2 − 4

17. ____________________________

18. x 4 − y8

18. ____________________________

19. 8 x3 + 27

19. ____________________________

20. 2c + 2d − cd − d 2

20. ____________________________

21. Find an equation whose graph has x-intercepts of 7 and –4.

21. ____________________________

For questions 22 and 23, solve. 22. 6 x 2 + x = 12

22. ____________________________

23. x3 + x 2 − 20 x = 0

23. ____________________________

24. A carpenter has a piece of wood 75 feet long. He wants to cut the piece of wood into three pieces, so that each consecutive piece is 2 feet longer than the previous one. Find the length of the shortest piece.

24. ____________________________

nt

⎛ c⎞ 25. The compound interest formula is A = P ⎜ 1 + ⎟ , where ⎝ n⎠ A is the amount, P is the principal invested, r is the annual rate of interest, n is the number of times the interest is compounded annually, and t is the time in years. Find the value of A if P = $3000, n = 3, r = 9%, and t = 1 year.

204

25. ____________________________

Chapter 5 Test Form E (cont.)

Name:

For questions 1–4, refer to the polynomial 5 − x 4 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

4. What is the leading coefficient of the polynomial?

4. ____________________________

For questions 5–10, perform the operation. 5.

(12 x

6.

(8a b − 4ab + 1) − ( 2ab

2

) (

− 8x + 6 − 5x2 + 7 + 2 x

2

2

(

2

7. − xy 2 7 x 2 y 2 + 2 xy 2 − 3 y

)

5. ____________________________

)

+ 2a 2 b + 1

6. ____________________________

)

7. ____________________________

8. (7 x + 2)(4 x − 3)

8. ____________________________

9. 3a(a + b)(a − b)

9. ____________________________

10.

(9x

2

)

+ 6 x − 4 ÷ ( 3x + 1)

10. ____________________________

If f ( x) = 10 x − 3 and g ( x) = x + 7 , find 11.

( f ⋅ g ) ( x)

11. ____________________________

12.

( f ⋅ g )( −1)

12. ____________________________

13. Use synthetic division to obtain the quotient

( 2x

3

13. ____________________________

)

+ 4 x − 2 x + 12 ÷ ( x + 3) . 2

14. Use the remainder theorem to find the remainder when 18 x3 + 33x 2 − 13x + 100 is divided by x + 3.

205

14. ____________________________

Name:

Chapter 5 Test Form E

Date:

For problems 15–20, factor completely. 15. x 2 − 5 x − 24

15. ____________________________

16. 20 x3 − 24 x5

16. ____________________________

17. ac + bd − ad − bc

17. ____________________________

18. 4 x 2 − 4 x − 3

18. ____________________________

19. 16 x3 y 2 − 12 x 2 y + 24 x3 y 3

19. ____________________________

20. 343z 3 − 27 w3

20. ____________________________

21. Find an equation whose graph has x-intercepts of –8 and 8.

21. ____________________________

For questions 22 and 23, solve. 22. 4 x 2 = 12 x

22. ____________________________

23. x 2 + 3x − 18 = 0

23. ____________________________

24. An area rug is rectangular. If the length of the rug is 2 feet shorter than twice its width, and the area of the rug is 60 square feet, find its length and width.

24. ____________________________

25. Find the area of the figure.

25. ____________________________

2 x — 8

x — 3

x

206

Chapter 5 Test Form F (cont.)

Name:

For questions 1–4, refer to the polynomial −8 + 3x5 − 2 x 2 . 1. Give the specific name of the polynomial.

1. ____________________________

2. Write the polynomial in descending powers of the variable x.

2. ____________________________

3. State the degree of the polynomial.

3. ____________________________

4. What is the leading coefficient of the polynomial?

4. ____________________________

For questions 5–10, perform the operation. 5.

(8 x

2

) (

− 4 x + 4 − −2 x 2 + 3 x − 7

(

6. − x 2 y −2 x 2 y 2 + xy + 2

)

5. ____________________________

)

6. ____________________________

7.

( 5 x + 3)( − x − 1)

7. ____________________________

8.

( 2 x − 8)( −4 x + 3)

8. ____________________________

9.

(1 − 5x )( − x

10.

2

( 24 x y

3 2

2

+ x −1

)

9. ____________________________

)

10. ____________________________

− 16 x 2 y + 3 ÷ 8 x 2 y

If f ( x) = 12 x − 20 and g ( x) = 2 x + 3 , find 11.

( f ⋅ g ) ( x)

11. ____________________________

12.

( f ⋅ g ) (2)

12. ____________________________

13. Use synthetic division to obtain the quotient ( x3 − 2 x 2 − 7 x + 24) ÷ ( x + 3).

13. ____________________________

14. Use the remainder theorem to find the remainder when 30 x 2 − 13x + 5 is divided by x + 1.

14. ____________________________

207

Name:

Chapter 5 Test Form F

Date:

For problems 15–20, factor completely. 15. x 2 − 7 x + 12

15. ____________________________

16. 16 x5 − 20 x 4 + 12 x3 + 20 x 2

16. ____________________________

17. 5 x 2 + 15 x + 2 x + 6

17. ____________________________

18. 8 x3 − 2 x 2 y − 3xy 2

18. ____________________________

19. 6 x3 − 11x 2 + 3x

19. ____________________________

20. 4r 2 − 81s 2

20. ____________________________

21. Find an equation whose graph has x intercepts of

1 and 2. 2

21. ____________________________

For questions 22 and 23, solve. 22. 9 x 2 − 6 x = 0

22. ____________________________

23. 4 x 2 + 9 x = 9

23. ____________________________

24. If each side of a square is increased by 7 meters, the area becomes 225 square meters. Find the length of a side of the original square.

24. ____________________________

25. A coconut is thrown upward with a speed of 48 feet per second from the top edge of a cliff 64 feet high. The height, h, of the coconut above the ground at any time, t, is given by the equation h = −16t 2 + 48t + 64 . Find the time it takes the coconut to hit the ground.

25. ____________________________

208

Chapter 5 Test Form G (cont.)

Name:

For questions 1–4, refer to the polynomial 7 − x 4 + 8 x 2 . 1. Give the specific name of the polynomial. (a) monomial

(b) binomial

(c) trinomial

(d) not polynomial

2. Write the polynomial in descending powers of the variable x. (a) 7 − x 4 + 8 x 2

(b) 7 + 8 x 2 − x 4

(c) − x 4 + 8 x 2 + 7

(d) 8 x 2 − x 4 + 7

(c) 6

(d) 8

(c) 7

(d) 8

(b) 5 x 2 + 5 x − 15

(c) 11x 2 + 3x + 9

(d) 11x 2 + 5 x − 15

(b) −14x3 y 3

(c) 14x 2 y 3

(d) 14x3 y 3

(b) −3x 2 − 10 x + 8

(c) 3x3 − 14 x + 8

(d) 3x 2 − 10 x + 8

(b) −2 y 3 + 5 y 2 − 10

(c) −2 y 3 + 10 y 2 − 20

(d) −10 y 5 + 20

3. State the degree of the polynomial. (a) 2

(b) 4

4. What is the leading coefficient of the polynomial? (a) –1

(b) 4

For questions 5–10, perform the operation. 5.

(8x

2

) (

+ 4 x − 3 − −3 x 2 − x + 12

(a) 5 x 2 + 3x + 9 6.

( −2 x )( −7 xy ) 2

3

(a) −14x 2 y3 7.

)

( −3x + 2 )( x + 4 ) (a) −3x 2 − 14 x + 8

(

)

8. −2 ⎡ y 3 − 5 y 2 − 2 ⎤ ⎣ ⎦ (a) −2 y 3 − 10 y 2 + 20 9.

( 24 x

2

(a) 8 x + 6 −

10.

(6x

4

)

+ 10 x − 8 ÷ ( 3 x − 1) 2 3x − 1

x−3 3x − 1

(b) 8 x + 6 −

14 3x − 1

(c) 8 x + 6

(d) 6 x + 5 +

(b) 2 x + 4 −

12

(c) 2 x + 4 y − 12

(d) 2 x + 4 y −

)

y + 12 x3 y − 36 ÷ 3 x3 y

(a) 2 x + 4 − 12

3

x y

209

12 x3 y 3

Name:

Chapter 5 Test Form G

Date:

If f ( x) = x − 5 and g ( x) = 4 x + 2 , find 11.

( f ⋅ g ) ( x) (a) 4 x 2 − 10

12.

(b) 4 x 2 + 8 x − 10

(c) 4 x 2 − 18 x − 10

(d) 5 x − 3

(b) 60

(c) 96

(d) 186

( f ⋅ g ) (7) (a) 32

(

)

13. Use synthetic division to obtain the quotient 3 x3 + 19 x 2 − 12 x + 14 ÷ ( x + 7 ) . (a) 3x 2 + 2 x + 2

(b) 3x 2 − 2 x + 2

(c) 3x 2 + 3x + 2

(d) 3x 2 − 3x + 2

14. Use the remainder theorem to find the remainder when 28 x 2 − 11x − 10 is divided by x – 2. (a) –14

(b) 124

(c) 5

(d) 80

(b) ( x + 3)( x − 2)

(c) ( x + 3)( x + 2)

(d) ( x − 5)( x + 6)

(b) (3 y + 4)( y − 1)

(c) (3 y − 4)( y + 1)

(d) (3 y + 2)( y − 2)

For problems 15–20, factor completely. 15. x 2 − 5 x + 6 (a) ( x − 3)( x − 2) 16. 3 y 2 − 4 y − 4 (a) (3 y − 2)( y + 2) 17. 4 x3 − 8 x 2 − 12 x

(

(a) x(4 x + 1)( x − 3)

(b) 4 x x 2 − 2 x − 3

)

(

)

(c) 4 x 2 + 1 ( x − 3)

(d) 4 x ( x + 1)( x − 3)

(c) (2 y + 5)(2 y − 5)

(d) (4 y + 25)(4 y − 25)

(x

)

18. 4 y 2 − 25 (a) (2 y − 5)2

(b) (2 y + 5)2

19. x6 + y 9 (a) (c)

(x (x

)( )⋅(x

2

− y3 ⋅ x4 + x2 y3 + y6

2

− y3

4

+ y6

)

)

(b)

2

(d)

)(

+ y3 ⋅ x 4 − 2 x2 y3 + y 6

(x

2

)(

+ y3 ⋅ x4 − x2 y3 + y6

20. 2 x3 y 3 + 250 x3

(

(a) 2 x3 ( y + 5) y 2 + 10 y + 25

)

( x ( y + 10) ⋅ ( 2 y

(b) 2 x3 ( y + 5) ⋅ y 2 − 5 y + 25

(c) x3 (2 y + 5) ⋅ ( y + 10)2

(d)

210

3

2

+ 25

)

)

)

Chapter 5 Test Form G (cont.) 21. Find an equation whose graph has x intercepts of (a) y = x 2 + 16 x − 4

Name:

1 and –4. 4

(b) y = x 2 − 16 x − 4

(c) y = 4 x 2 + 16 x − 4

(d) y = 4 x 2 + 15 x − 4

(b) x = 5 or x = 84

(c) x = 0 or x = −5

(d) x = –7 or x = 12

For questions 22 and 23, solve. 22. x( x − 5) = 84 (a) x = 0 or x = 5

23. 4 ( 9 x + 4 ) + 9 ( 9 x + 4 ) + 2 = 0 2

(a) x =

−2 5 or x = − 9 12

(b) x =

−2 −17 or x = 3 36

(c) x =

−2 −17 or x 9 36

(d) x =

−2 −5 or x = 3 12

24. The volume of a box is 1326 cubic feet. The width of the box is 6 feet and its height is 4 feet more than its length. Find the height of the box. (a) 13 ft

(b) 17 ft

(c) 9 ft

(d) 8 ft

(c) 12x 2 + x

(d) 12x 2 − x

25. Find the area of the shaded region in the figure. 4x

x

(a) 11x 2 + x

x — 1

3x

(b) 11x 2 − x

211

Chapter 5 Test Form H (cont.)

Name:

For questions 1–4, refer to the polynomial 12 − 10 x9 + 14 x11 . 1. Give the specific name of the polynomial. (a) monomial

(b) binomial

(c) trinomial

(d) not polynomial

2. Write the polynomial in descending powers of the variable x. (a) 12 − 10 x9 + 14 x11

(b) 12 + 14 x11 − 10 x9

(c) 14 x11 − 10 x9 + 12

(d) −10 x9 + 14 x11 + 12

(c) 14

(d) 20

(c) 10

(d) 14

(c) 4 x 2 y + 4 xy − 1

(d) 4 x 2 y + 4 xy −11

3. State the degree of the polynomial. (a) 9

(b) 11

4. What is the leading coefficient of the polynomial? (a) 9

(b) 12

For questions 5–10, perform the operation. 5.

(12 x

2

) (

y + 14 xy − 6 − 10 xy + 8 x 2 y − 5

(a) 2 x 2 y + 6 xy − 1

(b) 2 x 2 y + 22 xy −11

(

6. −a 2b3 −4a 2b3 + 2ab 2 − 3ab

7.

)

)

(a) 4a 4 b6 − 2a3b5 + 3a3b 4

(b) 4a 4 b 2 − 2a 2b6 + 3a 2 b3

(c) −4a 4b6 + 2a3b5 − 3a3b4

(d) −4a 4 b9 + 2a 2 b6 − 3a 2b3

( z + 4 )( z − 4 ) (a) z 2 + 16

(b) z 2 − 16

(c) z 2 + 8 z − 16

(d) z 2 − 8 z − 16

(b) 12 x 2 − 12 x − 21

(c) 12 x 2 + 19 x − 21

(d) 12 x 2 − 19 x − 21

8. (3x − 7)(4 x + 3) (a) 12 x 2 + 12 x − 21 9.

10.

( 3 − 6a )( −2a 2

2

+a+7

)

(a) −8a 2 + 3a + 21

(b) −6a 2 + 3a + 21

(c) 12a 4 − 6a3 − 48a 2 + 3a + 21 (d)

12a 4 − 6a3 − 36a 2 + 3a + 21

( 24 y

2

)

− 12 y + 5 ÷ ( 8 y + 1)

(a) 3 y + 1 +

7y + 4 8y +1

(b) 3 y − 1 −

4y 8y +1

(c) 3 y + 1 +

212

7 8y +1

(d) 3 y − 1 +

−7 y + 6 8y +1

Name:

Chapter 5 Test Form H

Date:

If f ( x) = 6 x + 3 and g ( x) = −2 x + 3 , find 11.

( f ⋅ g ) ( x) (a) −12 x 2 + 9

12.

(b) −12 x 2 − 12 x + 9

(c) −12 x 2 + 12 x + 9

(d) −12 x 2 + 14 x + 9

(b) –15

(c) –3

(d) 9

( f ⋅ g ) (−1) (a) –17

(

)

13. Use synthetic division to obtain the quotient 8 x 2 − 59 x + 21 ÷ ( x − 7 ) . (a) 8 x − 3

(b) 8 x + 3

(c) 4 x − 3

(d) 4 x + 3

14. Use the remainder theorem to find the remainder when 10 x 2 + 18 x + 3 is divided by x + 2. (a) 7

(b) 5

(c) 79

(d)

3 2

(d)

( 7 p − 6 )( 6 p − 1)

For problems 15–20, factor completely. 15. 42 p 2 + 29 p − 6 (a)

( 7 p + 6 )( 6 p − 1)

(b)

( 7 p + 6 )( 6 p + 1)

(c)

( 7 p − 6 )( 6 p + 1)

16. 8 x 2 − 8 x − 30 (a) −2 ( 2 x + 5 )( −2 x + 3)

(b) 2 ( 2 x + 5)( 2 x − 3)

(c) −2 ( 2 x + 3) ⋅ ( 2 x − 5)

(d) 2 ( 2 x − 5) ⋅ ( 2 x + 3)

17. x 4 − x 2 − 56 (a)

( x − 7 )( x + 8)

(b)

(x

)

(c)

( x + 7 )2 ( x − 8 )2

(d)

(x

(b)

( 4r + 3)( 4r − 3)

(c)

( 4r − 3 ) 2

(d)

( 4r − 9 )( 4r + 1)

(b)

( x − 8 )2

(c)

( x + 8 )2

(d)

(x

2

)(

− 7 x2 + 8

2

)(

+ 7 x2 − 8

)

18. 16r 2 + 24r + 9 (a)

( 4r + 3 ) 2

19. x 2 − 16 x + 64 (a)

( x − 8)( x + 8)

20. 8 x3 y − 34 x 2 y 2 + 8 xy 3

(

(a) 2 xy 4 x 2 + 15 xy − 4 y 2

)

(

(b) 2 xy 4 x 2 − 15 y − 4 y 2

(c) 2 xy ( 4 x + y ) ⋅ ( x − 4 y )

)

(d) 2 xy ( 4 x − y ) ⋅ ( x − 4 y )

213

2

+ 82

)

Chapter 5 Test Form H (cont.) 21. Find an equation whose graph has x intercepts of (a) y = 16 x 2 − 8 x + 3

Name:

1 −3 and . 4 4

(b) y = 16 x 2 + 8 x + 3

(c) y = 16 x 2 − 8 x − 3

(d) y = 16 x 2 + 8 x − 3

(b) x = –4 or x = −5

(c) x = –4 or x = 5

(d) x = 4 or x = 5

1 1 or x = 2 3

(d) x = 2 or x = 3

For questions 22 and 23, solve. 22. x 2 + x − 20 = 0 (a) x = 4 or x = −5 23. 6 x 2 − 5 x + 1 = 0 (a) x = -1 or x = 5

(b) x = −

1 1 or x = − 2 3

(c) x =

24. The volume of a box is 2240 cubic feet. The width of the box is 10 feet and its height is 2 feet more than its length. Find the height of the box. (a) 12 ft

(b) 16 ft

(c) 17 ft

(d) 14 ft

(c) x = 2, 4 or -6

(d) x = 2, -4 or 6

25. Find the x intercepts of the equation y = 2 x 2 + 4 x − 48 . (a) x = 4 or x = -6

(b) x = -4 or x = 6

214