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Chapter 6 Review Exercises

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Chapter Review Exercises

Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

12a^2b^3,\space 15ab^2

Answer

3ab^2

12m^2n^3,42m^5n^3

15y^3,\space 21y^2,\space 30y

Answer

3y

45x^3y^2,\space 15x^4y,\space 10x^5y^3

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

35y+84

Answer

7(5y+12)

6y^2+12y−6

18x^3−15x

Answer

3x(6x^2−5)

15m^4+6m^2n

4x^3−12x^2+16x

Answer

4x(x^2−3x+4)

−3x+24

−3x^3+27x^2−12x

Answer

−3x(x^2−9x+4)

3x(x−1)+5(x−1)

Factor by Grouping

In the following exercises, factor by grouping.

ax−ay+bx−by

Answer

(a+b)(x−y)

x^2y−xy^2+2x−2y

x^2+7x−3x−21

Answer

(x−3)(x+7)

4x^2−16x+3x−12

m^3+m^2+m+1

Answer

(m^2+1)(m+1)

5x−5y−y+x

Factor Trinomials

Factor Trinomials of the Form x^2+bx+c

In the following exercises, factor each trinomial of the form x^2+bx+c.

a^2+14a+33

Answer

(a+3)(a+11)

k^2−16k+60

m^2+3m−54

Answer

(m+9)(m−6)

x^2−3x−10

In the following examples, factor each trinomial of the form x^2+bxy+cy^2.

x^2+12xy+35y^2

Answer

(x+5y)(x+7y)

r^2+3rs−28s^2

a^2+4ab−21b^2

Answer

(a+7b)(a−3b)

p^2−5pq−36q^2

m^2−5mn+30n^2

Answer

Prime

Factor Trinomials of the Form ax2+bx+cax2+bx+c Using Trial and Error

In the following exercises, factor completely using trial and error.

x^3+5x^2−24x

3y^3−21y^2+30y

Answer

3y(y−5)(y−2)

5x^4+10x^3−75x^2

5y^2+14y+9

Answer

(5y+9)(y+1)

8x^2+25x+3

10y^2−53y−11

Answer

(5y+1)(2y−11)

6p^2−19pq+10q^2

−81a^2+153a+18

Answer

−9(9a−1)(a+2)

Factor Trinomials of the Form ax2+bx+cax2+bx+c using the ‘ac’ Method

In the following exercises, factor.

2x^2+9x+4

18a^2−9a+1

Answer

(3a−1)(6a−1)

15p^2+2p−8

15x^2+6x−2

Answer

(3x−1)(5x+2)

8a^2+32a+24

3x^2+3x−36

Answer

3(x+4)(x−3)

48y^2+12y−36

18a^2−57a−21

Answer

3(2a−7)(3a+1)

3n^4−12n^3−96n^2

Factor using substitution

In the following exercises, factor using substitution.

x^4−13x^2−30

Answer

(x^2−15)(x^2+2)

(x−3)^2−5(x−3)−36

Factor Special Products

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

25x^2+30x+9

Answer

(5x+3)^2

36a^2−84ab+49b^2

40x^2+360x+810

Answer

10(2x+9)^2

5k^3−70k^2+245k

75u^4−30u^3v+3u^2v^2

Answer

3u^2(5u−v)^2

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

81r^2−25

169m^2−n^2

Answer

(13m+n)(13m−n)

25p^2−1

9−121y^2

Answer

(3+11y)(3−11y)

20x^2−125

169n^3−n

Answer

n(13n+1)(13n−1)

6p^2q^2−54p^2

24p^2+54

Answer

6(4p^2+9)

49x^2−81y^2

16z^4−1

Answer

(2z−1)(2z+1)(4z^2+1)

48m^4n^2−243n^2

a^2+6a+9−9b^2

Answer

(a+3−3b)(a+3+3b)

x^2−16x+64−y^2

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

a^3−125

Answer

(a−5)(a^2+5a+25)

b^3−216

2m^3+54

Answer

2(m+3)(m^2−3m+9)

81m^3+3

General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

24x^3+44x^2

Answer

4x^2(6x+11)

24a^4−9a^3

16n^2−56mn+49m^2

Answer

(4n−7m)^2

6a^2−25a−9

5u^4−45u^2

Answer

5u^2(u+3)(u−3)

n^4−81

64j^2+225

Answer

prime

5x^2+5x−60

b^3−64

Answer

(b−4)(b^2+4b+16)

m^3+125

2b^2−2bc+5cb−5c^2

Answer

(2b+5c)(b−c)

48x^5y^2−243xy^2

5q^2−15q−90

Answer

5(q+3)(q−6)

4u^5v+4u^2v^3

10m^4−6250

Answer

10(m−5)(m+5)(m^2+25)

60x^2y−75xy+30y

16x^2−24xy+9y^2−64

Answer

(4x−3y+8)(4x−3y−8)

Polynomial Equations

Use the Zero Product Property

In the following exercises, solve.

(a−3)(a+7)=0

(5b+1)(6b+1)=0

Answer

b=−\frac{1}{5},\space b=−\frac{1}{6}

6m(12m−5)=0

(2x−1)^2=0

Answer

x=\frac{1}{2}

3m(2m−5)(m+6)=0

Solve Quadratic Equations by Factoring

In the following exercises, solve.

x^2+9x+20=0

Answer

x=−4,\space x=−5

y^2−y−72=0

2p^2−11p=40

Answer

p=−\frac{5}{2},p=8

q^3+3q^2+2q=0

144m^2−25=0

Answer

m=\frac{5}{12},\space m=−\frac{5}{12}

4n^2=36

(x+6)(x−3)=−8

Answer

x=2,\space x=−5

(3x−2)(x+4)=12

16p^3=24p^2+9p

Answer

p=0,\space p=\frac{3}{4}

2y^3+2y^2=12y

Solve Equations with Polynomial Functions

In the following exercises, solve.

For the function, f(x)=x^2+11x+20, ⓐ find when f(x)=−8 ⓑ Use this information to find two points that lie on the graph of the function.

Answer

x=−7 or \x=−4
(−7,−8) (−4,−8)

For the function, f(x)=9x^2−18x+5, ⓐ find when f(x)=−3 ⓑ Use this information to find two points that lie on the graph of the function.

In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

f(x)=64x^2−49

Answer

x=\frac{7}{8} or x=−\frac{7}{8}
(\frac{7}{8},0),\space (−\frac{7}{8},0)(0,−49)

f(x)=6x^2−13x−5

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

The product of two consecutive numbers is 399. Find the numbers.

Answer

The numbers are −21 and −19 or 19 and 21.

The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.

A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

Answer

The lengths are 8, 15, and 17 ft.

Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function h(t)=−16t^2+64t+80 models the height, h, of the ball above the ground as a function of time, t. Find: ⓐthe zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 80 feet above the ground. ⓒ the height the ball will be at t=2 seconds which is when the ball will be at its highest point.

Chapter Practice Test

In the following exercises, factor completely.

80a^2+120a^3

Answer

40a^2(2+3a)

5m(m−1)+3(m−1)

x^2+13x+36

Answer

(x+7)(x+6)

p^2+pq−12q^2

xy−8y+7x−56

Answer

(x−8)(y+7)

40r^2+810

9s^2−12s+4

Answer

(3s−2)^2

6x^2−11x−10

3x^2−75y^2

Answer

3(x+5y)(x−5y)

6u^2+3u−18

x^3+125

Answer

(x+5)(x^2−5x+25)

32x^5y^2−162xy^2

6x^4−19x^2+15

Answer

(3x^2−5)(2x^2−3)

3x^3−36x^2+108x

In the following exercises, solve

5a^2+26a=24

Answer

a=\frac{4}{5},\space a=−6

The product of two consecutive integers is 156. Find the integers.

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

Answer

The width is 12 inches and the length is 14 inches.

Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function h(t)=−16t^2+64t+80 models the height, h, of the ball above the ground as a function of time, t. Find: ⓐthe zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 128 feet above the ground. ⓒ the height the ball will be at t=4 seconds.

For the function, f(x)=x^2−7x+5, ⓐ find when f(x)=−7 ⓑ Use this information to find two points that lie on the graph of the function.

Answer

x=3 or x=4(3,−7) (4,−7)

For the function f(x)=25x^2−81, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒthe y-intercept of the graph of the function.

Glossary

degree of the polynomial equation
The degree of the polynomial equation is the degree of the polynomial.
polynomial equation
A polynomial equation is an equation that contains a polynomial expression.
quadratic equation
Polynomial equations of degree two are called quadratic equations.
zero of the function
A value of xx where the function is 0, is called a zero of the function.
Zero Product Property
The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.

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