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6.E: Review Exercises and Sample Exam

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Sep 2, 2024
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- 6.7: Applications Involving Quadratic Equations
- 7: Rational Expressions and Equations

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Review Exercises

Exercise $\PageIndex{1}$ Introduction to Factoring

Determine the missing factor.

$12x^{3}−24x^{2}+4x=4x( ? )$
$10y^{4}−35y^{3}−5y^{2}=5y^{2}( ? )$
$−18a^{5}+9a^{4}−27a^{3}=−9a^{3}( ? )$
$−21x^{2}y+7xy^{2}−49xy=−7xy( ? )$

Answer

1. $(3x^{2}−6x+1)$

3. $(2a^{2}−a+3)$

Exercise $\PageIndex{2}$ Introduction to Factoring

Factor out the GCF.

$22x^{2}+11x$
$15y^{4}−5y^{3}$
$18a^{3}−12a^{2}+30a$
$12a^{5}+20a^{3}−4a$
$9x^{3}y^{2}−18x^{2}y^{2}+27xy^{2}$
$16a^{5}b^{5}c−8a^{3}b^{6}+24a^{3}b^{2}c$

Answer

1. $11x(2x+1)$

3. $6a(3a^{2}−2a+5)$

5. $9xy2(x^{2}−2x+3)$

Exercise $\PageIndex{3}$ Introduction to Factoring

Factor by grouping.

$x^{2}+2x−5x−10$
$2x^{2}−2x−3x+3$
$x^{3}+5x^{2}−3x−15$
$x^{3}−6x^{2}+x−6$
$x^{3}−x^{2}y−2x+2y$
$a^{2}b^{2}−2a^{3}+6ab−3b^{3}$

Answer

1. $(x+2)(x−5)$

3. $(x+5)(x^{2}−3)$

5. $(x−y)(x^{2}−2)$

Exercise $\PageIndex{4}$ Factoring Trinomials of the Form $x^{2}+bx+c$

Are the following factored correctly? Check by multiplying.

$x^{2}+5x+6=(x+6)(x−1)$
$x^{2}+3x−10=(x+5)(x−2)$
$x^{2}+6x+9=(x+3)^{2}$
$x^{2}−6x−9=(x−3)(x+3)$

Answer

1. No

3. Yes

Exercise $\PageIndex{5}$ Factoring Trinomials of the Form $x^{2}+bx+c$

Factor.

$x^{2}−13x−14$
$x^{2}+13x+12$
$y^{2}+10y+25$
$y^{2}−20y+100$
$a^{2}−8a−48$
$b^{2}−18b+45$
$x^{2}+2x+24$
$x^{2}−10x−16$
$a^{2}+ab−2b^{2}$
$a^{2}b^{2}+5ab−50$

Answer

1. $(x−14)(x+1)$

3. $(y+5)^{2}$

5. $(a−12)(a+4)$

7. Prime

9. $(a−b)(a+2b)$

Exercise $\PageIndex{6}$ Factoring Trinomials of the Form $ax^{2}+bx+c$

Factor.

$5x^{2}−27x−18$
$3x^{2}−14x+8$
$4x^{2}−28x+49$
$9x^{2}+48x+64$
$6x^{2}−29x−9$
$8x^{2}+6x+9$
$60x^{2}−65x+15$
$16x^{2}−40x+16$
$6x^{3}−10x^{2}y+4xy^{2}$
$10x^{3}y−82x^{2}y^{2}+16xy^{3}$
$−y^{2}+9y+36$
$−a^{2}−7a+98$
$16+142x−18x^{2}$
$45−132x−60x^{2}$

Answer

1. $(5x+3)(x−6)$

3. $(2x−7)^{2}$

5. Prime

7. $5(3x−1)(4x−3)$

9. $2x(3x−2y)(x−y)$

11. $−1(y−12)(y+3)$

13. $−2(9x+1)(x−8)$

Exercise $\PageIndex{7}$ Factoring Special Binomials

Factor completely.

$x^{2}−81$
$25x^{2}−36$
$4x^{2}−49$
$81x^{2}−1$
$x^{2}−64y^{2}$
$100x^{2}y^{2}−1$
$16x^{4}−y^{4}$
$x^{4}−81y^{4}$
$8x^{3}−125$
$27+y^{3}$
$54x^{4}y−2xy^{4}$
$3x^{4}y^{2}+24xy^{5}$
$64x^{6}−y^{6}$
$x^{6}+1$

Answer

1. $(x+9)(x−9)$

3. $(2x+7)(2x−7)$

5. $(x+8y)(x−8y)$

7. $(4x^{2}+y^{2})(2x+y)(2x−y)$

9. $(2x−5)(4x^{2}+10x+25)$

11. $2xy(3x−y)(9x^{2}+3xy+y^{2})$

13. $(2x+y)(4x^{2}−2xy+y^{2})(2x−y)(4x^{2}+2xy+y^{2})$

Exercise $\PageIndex{8}$ General Guidelines for Factoring Polynomials

Factor completely.

$8x^{3}−4x^{2}+20x$
$50a^{4}b^{4}c+5a^{3}b^{5}c^{2}$
$x^{3}−12x^{2}−x+12$
$a^{3}−2a^{2}−3ab+6b$
$−y^{2}−15y+16$
$x^{2}−18x+72$
$144x^{2}−25$
$3x^{4}−48$
$20x^{2}−41x−9$
$24x^{2}+14x−20$
$a^{4}b−343ab^{4}$
$32x^{7}y^{2}+4xy^{8}$

Answer

1. $4x(2x^{2}−x+5)$

3. $(x−12)(x+1)(x−1)$

5. $−1(y+16)(y−1)$

7. $(12x+5)(12x−5)$

9. $(4x−9)(5x+1)$

11. $ab(a−7b)(a^{2}+7ab+49b^{2})$

Exercise $\PageIndex{9}$ Solving Equations by Factoring

Solve.

$(x−9)(x+10)=0$
$−3x(x+8)=0$
$6(x+1)(x−1)=0$
$(x−12)(x+4)(2x−1)=0$
$x^{2}+5x−50=0$
$3x^{2}−13x+4=0$
$3x^{2}−12=0$
$16x^{2}−9=0$
$(x−2)(x+6)=20$
$2(x−2)(x+3)=7x−9$
$52x^{2}−203x=0$
$23x^{2}−512x+124=0$

Answer

1. $9, −10$

3. $−1, 1$

5. $−10, 5$

7. $±2$

9. $−8, 4$

11. $0, \frac{8}{3}$

Exercise $\PageIndex{10}$ Solving Equations by Factoring

Find a quadratic equation with integer coefficients, given the following solutions.

$−7, 6$
$0, −10$
$−\frac{1}{9}, \frac{1}{2}$
$± \frac{3}{2}$

Answer

1. $x^{2}+x−42=0$

3. $18x^{2}−7x−1=0$

Exercise $\PageIndex{11}$ Applications Involving Quadratic Equations

Set up an algebraic equation and then solve the following.

An integer is $4$ less than twice another. If the product of the two integers is $96$ , then find the integers.
The sum of the squares of two consecutive positive even integers is $52$ . Find the integers.
A $20$ -foot ladder leaning against a wall reaches a height that is $4$ feet more than the distance from the wall to the base of the ladder. How high does the ladder reach?
The height of an object dropped from the top of a $196$ -foot building is given by $h(t)=−16t^{2}+196$ , where $t$ represents the number of seconds after the object has been released. How long will it take the object to hit the ground?
The length of a rectangle is $1$ centimeter less than three times the width. If the area is $70$ square centimeters, then find the dimensions of the rectangle.
The base of a triangle is $4$ centimeters more than twice the height. If the area of the triangle is $80$ square centimeters, then find the measure of the base.

Answer

1. { $8, 12$ } or { $−6, −16$ }

3. $16$ feet

5. Length: $14$ centimeters; width: $5$ centimeters

Sample Exam

Exercise $\PageIndex{12}$

Determine the GCF of the terms $25a^{2}b^{2}c, 50ab^{4}$ , and $35a^{3}b^{3}c^{2}$ .
Determine the missing factor: $24x^{2}y^{3}−16x^{3}y^{2}+8x^{2}y=8x^{2}y( ? )$ .

Answer: 1. $5ab^{2}$

Exercise $\PageIndex{13}$

Factor.

$12x^{5}−15x^{4}+3x^{2}$
$x^{3}−4x^{2}−2x+8$
$x^{2}−7x+12$
$9x^{2}−12x+4$
$x^{2}−81$
$x^{3}+27y^{3}$

Answer

1. $3x^{2}(4x^{3}−5x^{2}+1)$

3. $(x−4)(x−3)$

5. $(x+9)(x−9)$

Exercise $\PageIndex{14}$

Factor completely.

$x^{3}+2x^{2}−4x−8$
$x^{4}−1$
$−6x^{3}+20x^{2}−6x$
$x^{6}−1$

Answer

1. $(x+2)^{2}(x−2)$

3. $−2x(3x−1)(x−3)$

Exercise $\PageIndex{15}$

Solve.

$(2x+1)(x−7)=0$
$3x(4x−3)(x+1)=0$
$x^{2}−64=0$
$x^{2}+4x−12=0$
$23x^{2}+89x−16=0$
$(x−5)(x−3)=−1$
$3x(x+3)=14x+2$
$(3x+1)(3x+2)=9x+3$

Answer

1. $−\frac{1}{2}, 7$

3. $±8$

5. $−\frac{3}{2}, \frac{1}{6}$

7. $−\frac{1}{3}, 2$

Exercise $\PageIndex{16}$

For each problem, set up an algebraic equation and then solve.

An integer is $4$ less than twice another. If the product of the two integers is $70$ , then find the integers.
The sum of the squares of two consecutive positive odd integers is $130$ . Find the integers.
The length of a rectangle is $4$ feet more than twice its width. If the area is $160$ square feet, then find the dimensions of the rectangle.
The height of a triangle is $6$ centimeters less than four times the length of its base. If the area measures $27$ square centimeters, then what is the height of the triangle?
The height of a projectile launched upward at a speed of $64$ feet/second from a height of $36$ feet is given by the function $h(t)=−16t^{2}+64t+36$ . How long will it take the projectile to hit the ground?

Answer

1. { $7, 10$ } or { $−14, −5$ }

3. Width: $8$ feet; length: $20$ feet

5. $4\frac{1}{2}$ sec

Review Exercises

Exercise 6.E.1\PageIndex{1} Introduction to Factoring

Exercise \PageIndex{2}\PageIndex{2} Introduction to Factoring

Exercise \PageIndex{3}\PageIndex{3} Introduction to Factoring

Exercise \PageIndex{4}\PageIndex{4} Factoring Trinomials of the Form x^{2}+bx+cx^{2}+bx+c

Exercise \PageIndex{5}\PageIndex{5} Factoring Trinomials of the Form x^{2}+bx+cx^{2}+bx+c

Exercise \PageIndex{6}\PageIndex{6} Factoring Trinomials of the Form ax^{2}+bx+cax^{2}+bx+c

Exercise \PageIndex{7}\PageIndex{7} Factoring Special Binomials

Exercise \PageIndex{8}\PageIndex{8} General Guidelines for Factoring Polynomials

Exercise \PageIndex{9}\PageIndex{9} Solving Equations by Factoring

Exercise \PageIndex{10}\PageIndex{10} Solving Equations by Factoring

Exercise \PageIndex{11}\PageIndex{11} Applications Involving Quadratic Equations

Sample Exam

Exercise \PageIndex{12}\PageIndex{12}

Exercise \PageIndex{13}\PageIndex{13}

Exercise \PageIndex{14}\PageIndex{14}

Exercise \PageIndex{15}\PageIndex{15}

Exercise \PageIndex{16}\PageIndex{16}

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Exercise $\PageIndex{1}$ Introduction to Factoring

Exercise $\PageIndex{2}$ Introduction to Factoring

Exercise $\PageIndex{3}$ Introduction to Factoring

Exercise $\PageIndex{4}$ Factoring Trinomials of the Form $x^{2}+bx+c$

Exercise $\PageIndex{5}$ Factoring Trinomials of the Form $x^{2}+bx+c$

Exercise $\PageIndex{6}$ Factoring Trinomials of the Form $ax^{2}+bx+c$

Exercise $\PageIndex{7}$ Factoring Special Binomials

Exercise $\PageIndex{8}$ General Guidelines for Factoring Polynomials

Exercise $\PageIndex{9}$ Solving Equations by Factoring

Exercise $\PageIndex{10}$ Solving Equations by Factoring

Exercise $\PageIndex{11}$ Applications Involving Quadratic Equations

Exercise $\PageIndex{12}$

Exercise $\PageIndex{13}$

Exercise $\PageIndex{14}$

Exercise $\PageIndex{15}$

Exercise $\PageIndex{16}$