Question 1

The binomial $4 x^{2}+16$ is the sum of two squares that can be factored. Why is this possible?

Accepted Answer

The binomial has a c

Question 2

Find all solutions by factoring.
-$(5 \mathrm{x}+16)(\mathrm{x}+3)=(4 \mathrm{x}+14)(\mathrm{x}+2)$

Accepted Answer

A)  $\left\{-2,-\frac{7}{2}, 1\right\}$ 
B)  $\left\{-\frac{16}{5},-3,-\frac{7}{2},-2\right\}$ 
C)  $\{-3,-2\}$ 
D)  $\{-5,-4\}$ 
A)  $\left\{-2,-\frac{7}{2}, 1\right\}$ 
B)  $\left\{-\frac{16}{5},-3,-\frac{7}{2},-2\right\}$ 
C)  $\{-3,-2\}$ 
D)  $\{-5,-4\}$

Question 3

$x^{2}+4 x+4-9 z^{2}$

Accepted Answer

A)  $(x+2-3 z)^{2}$ 
B)  $(x+2-3 z)(x+2+3 z)$ 
C)  $(x-2-3 z)(x+2+3 z)$ 
D)  $(3 z(x+2))^{2}$ 
A)  $(x+2-3 z)^{2}$ 
B)  $(x+2-3 z)(x+2+3 z)$ 
C)  $(x-2-3 z)(x+2+3 z)$ 
D)  $(3 z(x+2))^{2}$

Question 4

Factor the trinomial completely.
-$-25 x^{2}+30 x+16$

Accepted Answer

A)  $-(5 x+2)(5 x-8)$ 
B)  $(5 x+2)(5 x+8)$ 
C)  $-(5 x-2)(5 x-8)$ 
D)  $-(5 x-2)(5 x+8)$ 
A)  $-(5 x+2)(5 x-8)$ 
B)  $(5 x+2)(5 x+8)$ 
C)  $-(5 x-2)(5 x-8)$ 
D)  $-(5 x-2)(5 x+8)$

Question 5

$8 z^{2}-16 z$

Accepted Answer

A)  $8 z(z-2)$ 
B)  $8\left(z^{2}-2 z\right)$ 
C)  $2 z(8 z-1)$ 
D)  $8 z(z-8)$ 
A)  $8 z(z-2)$ 
B)  $8\left(z^{2}-2 z\right)$ 
C)  $2 z(8 z-1)$ 
D)  $8 z(z-8)$

Question 6

Solve the equation for the specified variable.
-$c=\frac{7 t+6}{t}$ for $\mathrm{t}$

Accepted Answer

A)  $\mathrm{t}=\frac{-6}{\mathrm{c}-7}$ or $\mathrm{t}=\frac{6}{-\mathrm{c}+7}$ 
B)  $\mathrm{t}=\frac{13}{\mathrm{c}}$ or $\mathrm{t}=\frac{-13}{-\mathrm{c}}$ 
C)  $\mathrm{t}=\frac{6}{\mathrm{c}-7}$ or $\mathrm{t}=\frac{-6}{-\mathrm{c}+7}$ 
D)  $\mathrm{t}=\frac{\mathrm{c}+7}{6}$ or $\mathrm{t}=\frac{-\mathrm{c}-7}{-6}$ 
A)  $\mathrm{t}=\frac{-6}{\mathrm{c}-7}$ or $\mathrm{t}=\frac{6}{-\mathrm{c}+7}$ 
B)  $\mathrm{t}=\frac{13}{\mathrm{c}}$ or $\mathrm{t}=\frac{-13}{-\mathrm{c}}$ 
C)  $\mathrm{t}=\frac{6}{\mathrm{c}-7}$ or $\mathrm{t}=\frac{-6}{-\mathrm{c}+7}$ 
D)  $\mathrm{t}=\frac{\mathrm{c}+7}{6}$ or $\mathrm{t}=\frac{-\mathrm{c}-7}{-6}$

Question 7

Solve the equation for the specified variable.
-$-7 \mathrm{~s}+8 \mathrm{p}=\mathrm{tp}-8$ for $\mathrm{p}$

Accepted Answer

A)  $\mathrm{p}=\frac{-7 \mathrm{~s}+8}{8}$ or $\mathrm{p}=\frac{7 \mathrm{~s}-8}{-8}$ 
B)  $\mathrm{p}=\frac{-7 \mathrm{~s}+8}{-\mathrm{t}}$ or $\mathrm{p}=\frac{7 \mathrm{~s}-8}{\mathrm{t}}$ 
C)  $p=\frac{7 s-8}{8-t}$ or $p=\frac{-7 s+8}{t-8}$ 
D)  $\mathrm{p}=\frac{8-\mathrm{t}}{7 \mathrm{~s}-8}$ or $\mathrm{p}=\frac{\mathrm{t}-8}{-7 \mathrm{~s}+8}$ 
A)  $\mathrm{p}=\frac{-7 \mathrm{~s}+8}{8}$ or $\mathrm{p}=\frac{7 \mathrm{~s}-8}{-8}$ 
B)  $\mathrm{p}=\frac{-7 \mathrm{~s}+8}{-\mathrm{t}}$ or $\mathrm{p}=\frac{7 \mathrm{~s}-8}{\mathrm{t}}$ 
C)  $p=\frac{7 s-8}{8-t}$ or $p=\frac{-7 s+8}{t-8}$ 
D)  $\mathrm{p}=\frac{8-\mathrm{t}}{7 \mathrm{~s}-8}$ or $\mathrm{p}=\frac{\mathrm{t}-8}{-7 \mathrm{~s}+8}$

Question 8

Solve the equation.
-$(x+3)(x-4)(x-18)=0$

Accepted Answer

A)  $\{3,4,18\}$ 
B)  $\{-3,4,18\}$ 
C)  $\{-3,4,-18\}$ 
D)  $\{0,-3,4\}$ 
A)  $\{3,4,18\}$ 
B)  $\{-3,4,18\}$ 
C)  $\{-3,4,-18\}$ 
D)  $\{0,-3,4\}$

Question 9

Factor the trinomial completely.
-$s^{2} t^{2}-7 s t+47$

Accepted Answer

A)  $(s t-47)(s t-1)$ 
B)  $(s t+47)(s t-1)$ 
C)  Prime 
D)  $(s t-47)(s t+1)$ 
A)  $(s t-47)(s t-1)$ 
B)  $(s t+47)(s t-1)$ 
C)  Prime 
D)  $(s t-47)(s t+1)$

Question 10

Factor the trinomial completely.
-$10 z^{2}-11 z-6$

Accepted Answer

A)  $(10 z-3)(z+2)$ 
B)  $(10 z-3)(z-6)$ 
C)  $(2 z-3)(5 z+2)$ 
D)  $(2 z+3)(5 z-2)$ 
A)  $(10 z-3)(z+2)$ 
B)  $(10 z-3)(z-6)$ 
C)  $(2 z-3)(5 z+2)$ 
D)  $(2 z+3)(5 z-2)$

Question 11

Solve the equation for the specified variable.
-$-3 k+a r=r-2 y$ for $r$

Accepted Answer

A)  $r=\frac{3 k-2 y}{a-1}$ or $r=\frac{-3 k+2 y}{1-a}$ 
B)  $r=\frac{-3 k+2 y}{a-1}$ or $r=\frac{3 k-2 y}{1-a}$ 
C)  $r=\frac{a-1}{3 k-2 y}$ or $r=\frac{1-a}{-3 k+2 y}$ 
D)  $r=\frac{-3 k+a}{1-2 y}$ or $r=\frac{3 k-a}{2 y-1}$ 
A)  $r=\frac{3 k-2 y}{a-1}$ or $r=\frac{-3 k+2 y}{1-a}$ 
B)  $r=\frac{-3 k+2 y}{a-1}$ or $r=\frac{3 k-2 y}{1-a}$ 
C)  $r=\frac{a-1}{3 k-2 y}$ or $r=\frac{1-a}{-3 k+2 y}$ 
D)  $r=\frac{-3 k+a}{1-2 y}$ or $r=\frac{3 k-a}{2 y-1}$

Question 12

Factor the polynomial completely
-$12 z^{6}+10 z^{4}-12 z^{2}$

Accepted Answer

A)  $\left(6 z^{3}-4\right)\left(2 z^{3}+3\right)$ 
B)  $2\left(3 z^{4}-2\right)\left(2 z^{2}+3\right)$ 
C)  $2 z^{2}\left(2 z^{2}-3\right)\left(3 z^{2}+2\right)$ 
D)  $2 z^{2}\left(3 z^{2}-2\right)\left(2 z^{2}+3\right)$ 
A)  $\left(6 z^{3}-4\right)\left(2 z^{3}+3\right)$ 
B)  $2\left(3 z^{4}-2\right)\left(2 z^{2}+3\right)$ 
C)  $2 z^{2}\left(2 z^{2}-3\right)\left(3 z^{2}+2\right)$ 
D)  $2 z^{2}\left(3 z^{2}-2\right)\left(2 z^{2}+3\right)$

Question 13

Solve the equation
-$3(3 z-4)^{2}-7(3 z-4)+2=0$

Accepted Answer

A)  $\left\{-\frac{2}{3},-\frac{11}{9}\right\}$ 
B)  $\left\{\frac{2}{3}, \frac{13}{3}\right\}$ 
C)  $\left\{\frac{1}{3}, 2\right\}$ 
D)  $\left\{2, \frac{13}{9}\right\}$ 
A)  $\left\{-\frac{2}{3},-\frac{11}{9}\right\}$ 
B)  $\left\{\frac{2}{3}, \frac{13}{3}\right\}$ 
C)  $\left\{\frac{1}{3}, 2\right\}$ 
D)  $\left\{2, \frac{13}{9}\right\}$

Question 14

$a^{3}+7 a^{2}-a b^{2}-7 b^{2}$

Accepted Answer

A)  $(a+b)(a-b)(a+7)$ 
B)  $(a-7)\left(a^{2}+b^{2}\right)$ 
C)  $(a+b)^{2}(a+7)$ 
D)  $\left(a^{2}+b^{2}\right)(a+7)$ 
A)  $(a+b)(a-b)(a+7)$ 
B)  $(a-7)\left(a^{2}+b^{2}\right)$ 
C)  $(a+b)^{2}(a+7)$ 
D)  $\left(a^{2}+b^{2}\right)(a+7)$

Question 15

Factor the polynomial completely
-$24 k^{3} m-81 m^{4}$

Accepted Answer

A)  $3 m(2 k-3 m)\left(4 k^{2}+6 k m+9 m^{2}\right)$ 
B)  $3 m(8 k-3 m)\left(k^{2}+6 k m+9 m^{2}\right)$ 
C)  $\left(6 \mathrm{~km}-9 \mathrm{~m}^{2}\right)\left(4 \mathrm{k}^{2}+9 \mathrm{~m}^{2}\right)$ 
D)  $3 \mathrm{~m}\left(2 \mathrm{k}+3 \mathrm{~m}^{2}\right)\left(4 \mathrm{k}^{2}-6 \mathrm{~km}+9 \mathrm{~km}^{2}\right)$ 
A)  $3 m(2 k-3 m)\left(4 k^{2}+6 k m+9 m^{2}\right)$ 
B)  $3 m(8 k-3 m)\left(k^{2}+6 k m+9 m^{2}\right)$ 
C)  $\left(6 \mathrm{~km}-9 \mathrm{~m}^{2}\right)\left(4 \mathrm{k}^{2}+9 \mathrm{~m}^{2}\right)$ 
D)  $3 \mathrm{~m}\left(2 \mathrm{k}+3 \mathrm{~m}^{2}\right)\left(4 \mathrm{k}^{2}-6 \mathrm{~km}+9 \mathrm{~km}^{2}\right)$

Question 16

actor out the greatest common factor. Simplify the factors, if possible
-$t(3-m)+s(3-m)$

Accepted Answer

A)  $(\mathrm{t}+\mathrm{s})(3+\mathrm{m})$ 
B)  $\mathrm{t}(3-\mathrm{m})+\mathrm{s}$ 
C)  $(\mathrm{t}-\mathrm{s})(3-\mathrm{m})$ 
D)  $(\mathrm{t}+\mathrm{s})(3-\mathrm{m})$ 
A)  $(\mathrm{t}+\mathrm{s})(3+\mathrm{m})$ 
B)  $\mathrm{t}(3-\mathrm{m})+\mathrm{s}$ 
C)  $(\mathrm{t}-\mathrm{s})(3-\mathrm{m})$ 
D)  $(\mathrm{t}+\mathrm{s})(3-\mathrm{m})$

Question 17

What values of $n$ will make $x^{n}$ a perfect square?

Accepted Answer

A)  $n$ must be odd. 
B)  $n$ must be a power of 3 . 
C)  $n$ must be even. 
D)  $n$ must be a multiple of 5 . 
A)  $n$ must be odd. 
B)  $n$ must be a power of 3 . 
C)  $n$ must be even. 
D)  $n$ must be a multiple of 5 .

Question 18

$7 \mathrm{~m}(\mathrm{~m}-4)=4(4-\mathrm{m})$

Accepted Answer

A)  $\left\{-\frac{4}{7}, 4\right\}$ 
B)  $\{0,-3\}$ 
C)  $\left\{\frac{4}{7}\right\}$ 
D)  $\left\{\frac{7}{4}, 4\right\}$ 
A)  $\left\{-\frac{4}{7}, 4\right\}$ 
B)  $\{0,-3\}$ 
C)  $\left\{\frac{4}{7}\right\}$ 
D)  $\left\{\frac{7}{4}, 4\right\}$

Question 19

Factor the polynomial
-$3 c^{2}+6 c d+3 d^{2}-75 f^{2}$

Accepted Answer

A)  $3(c+d+5 f)(c+d-5 f)$ 
B)  $(c+d+5 f)(c+d-5 f)$ 
C)  $\left((c+d)^{2}+5 f\right)\left((c+d)^{2}-5 f\right)$ 
D)  $3\left((c+d)^{2}+5 f\right)\left((c+d)^{2}-5 f\right)$ 
A)  $3(c+d+5 f)(c+d-5 f)$ 
B)  $(c+d+5 f)(c+d-5 f)$ 
C)  $\left((c+d)^{2}+5 f\right)\left((c+d)^{2}-5 f\right)$ 
D)  $3\left((c+d)^{2}+5 f\right)\left((c+d)^{2}-5 f\right)$

Question 20

Factor the polynomial completely
-$216 s^{3}+1$

Accepted Answer

A)  $(6 s+1)\left(36 s^{2}+1\right)$ 
B)  $(216 s+1)\left(s^{2}-6 s+1\right)$ 
C)  $(6 s-1)\left(36 s^{2}+6 s+1\right)$ 
D)  $(6 s+1)\left(36 s^{2}-6 s+1\right)$ 
A)  $(6 s+1)\left(36 s^{2}+1\right)$ 
B)  $(216 s+1)\left(s^{2}-6 s+1\right)$ 
C)  $(6 s-1)\left(36 s^{2}+6 s+1\right)$ 
D)  $(6 s+1)\left(36 s^{2}-6 s+1\right)$

The binomial $4 x^{2}+16$ is the sum of two squares that can be factored. Why is this possible?

Find all solutions by factoring. - $(5 \mathrm{x}+16)(\mathrm{x}+3)=(4 \mathrm{x}+14)(\mathrm{x}+2)$

$x^{2}+4 x+4-9 z^{2}$

Factor the trinomial completely. - $-25 x^{2}+30 x+16$

$8 z^{2}-16 z$

Solve the equation for the specified variable. - $c=\frac{7 t+6}{t}$ for $\mathrm{t}$

Solve the equation for the specified variable. - $-7 \mathrm{~s}+8 \mathrm{p}=\mathrm{tp}-8$ for $\mathrm{p}$

Solve the equation. - $(x+3)(x-4)(x-18)=0$

Factor the trinomial completely. - $s^{2} t^{2}-7 s t+47$

Factor the trinomial completely. - $10 z^{2}-11 z-6$

Solve the equation for the specified variable. - $-3 k+a r=r-2 y$ for $r$

Factor the polynomial completely - $12 z^{6}+10 z^{4}-12 z^{2}$

Solve the equation - $3(3 z-4)^{2}-7(3 z-4)+2=0$

$a^{3}+7 a^{2}-a b^{2}-7 b^{2}$

Factor the polynomial completely - $24 k^{3} m-81 m^{4}$

actor out the greatest common factor. Simplify the factors, if possible - $t(3-m)+s(3-m)$

What values of $n$ will make $x^{n}$ a perfect square?

$7 \mathrm{~m}(\mathrm{~m}-4)=4(4-\mathrm{m})$

Factor the polynomial - $3 c^{2}+6 c d+3 d^{2}-75 f^{2}$

Factor the polynomial completely - $216 s^{3}+1$

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Rational Expressions and Functions

Roots, Radicals, and Root Functions

Quadratic Equations, Inequalities, and Functions

Inverse, Exponential, and Logarithmic Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

Exam 6: Factoring

The binomial 4x2+164 x^{2}+164x2+16 is the sum of two squares that can be factored. Why is this possible?

Find all solutions by factoring. - (5x+16)(x+3)=(4x+14)(x+2)(5 \mathrm{x}+16)(\mathrm{x}+3)=(4 \mathrm{x}+14)(\mathrm{x}+2)(5x+16)(x+3)=(4x+14)(x+2)

x2+4x+4−9z2x^{2}+4 x+4-9 z^{2}x2+4x+4−9z2

Factor the trinomial completely. - −25x2+30x+16-25 x^{2}+30 x+16−25x2+30x+16

8z2−16z8 z^{2}-16 z8z2−16z

Solve the equation for the specified variable. - c=7t+6tc=\frac{7 t+6}{t}c=t7t+6​ for t\mathrm{t}t

Solve the equation for the specified variable. - −7 s+8p=tp−8-7 \mathrm{~s}+8 \mathrm{p}=\mathrm{tp}-8−7 s+8p=tp−8 for p\mathrm{p}p

Solve the equation. - (x+3)(x−4)(x−18)=0(x+3)(x-4)(x-18)=0(x+3)(x−4)(x−18)=0

Factor the trinomial completely. - s2t2−7st+47s^{2} t^{2}-7 s t+47s2t2−7st+47

Factor the trinomial completely. - 10z2−11z−610 z^{2}-11 z-610z2−11z−6

Solve the equation for the specified variable. - −3k+ar=r−2y-3 k+a r=r-2 y−3k+ar=r−2y for rrr

Factor the polynomial completely - 12z6+10z4−12z212 z^{6}+10 z^{4}-12 z^{2}12z6+10z4−12z2

Solve the equation - 3(3z−4)2−7(3z−4)+2=03(3 z-4)^{2}-7(3 z-4)+2=03(3z−4)2−7(3z−4)+2=0

a3+7a2−ab2−7b2a^{3}+7 a^{2}-a b^{2}-7 b^{2}a3+7a2−ab2−7b2

Factor the polynomial completely - 24k3m−81m424 k^{3} m-81 m^{4}24k3m−81m4

actor out the greatest common factor. Simplify the factors, if possible - t(3−m)+s(3−m)t(3-m)+s(3-m)t(3−m)+s(3−m)

What values of nnn will make xnx^{n}xn a perfect square?

7 m( m−4)=4(4−m)7 \mathrm{~m}(\mathrm{~m}-4)=4(4-\mathrm{m})7 m( m−4)=4(4−m)

Factor the polynomial - 3c2+6cd+3d2−75f23 c^{2}+6 c d+3 d^{2}-75 f^{2}3c2+6cd+3d2−75f2

Factor the polynomial completely - 216s3+1216 s^{3}+1216s3+1

Review of the Real Number System

Linear Equations, Inequalities, and Applications

Linear Equations, Graphs, and Functions

Systems of Linear Equations

Exponents, Polynomials, and Polynomial Functions

Rational Expressions and Functions

Roots, Radicals, and Root Functions

Quadratic Equations, Inequalities, and Functions

Inverse, Exponential, and Logarithmic Functions

Nonlinear Functions, Conic Sections, and Nonlinear Systems

Further Topics in Algebra

Appendices

Filters

The binomial $4 x^{2}+16$ is the sum of two squares that can be factored. Why is this possible?

Find all solutions by factoring. - $(5 \mathrm{x}+16)(\mathrm{x}+3)=(4 \mathrm{x}+14)(\mathrm{x}+2)$

$x^{2}+4 x+4-9 z^{2}$

Factor the trinomial completely. - $-25 x^{2}+30 x+16$

$8 z^{2}-16 z$

Solve the equation for the specified variable. - $c=\frac{7 t+6}{t}$ for $\mathrm{t}$

Solve the equation for the specified variable. - $-7 \mathrm{~s}+8 \mathrm{p}=\mathrm{tp}-8$ for $\mathrm{p}$

Solve the equation. - $(x+3)(x-4)(x-18)=0$

Factor the trinomial completely. - $s^{2} t^{2}-7 s t+47$

Factor the trinomial completely. - $10 z^{2}-11 z-6$

Solve the equation for the specified variable. - $-3 k+a r=r-2 y$ for $r$

Factor the polynomial completely - $12 z^{6}+10 z^{4}-12 z^{2}$

Solve the equation - $3(3 z-4)^{2}-7(3 z-4)+2=0$

$a^{3}+7 a^{2}-a b^{2}-7 b^{2}$

Factor the polynomial completely - $24 k^{3} m-81 m^{4}$

actor out the greatest common factor. Simplify the factors, if possible - $t(3-m)+s(3-m)$

What values of $n$ will make $x^{n}$ a perfect square?

$7 \mathrm{~m}(\mathrm{~m}-4)=4(4-\mathrm{m})$

Factor the polynomial - $3 c^{2}+6 c d+3 d^{2}-75 f^{2}$

Factor the polynomial completely - $216 s^{3}+1$