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College Algebra : College Algebra

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Example Questions

Example Question #21 : Simplifying Polynomials

Divide:

$\left (x^{2} + 6x -7 \right )\div (x+3)$

Possible Answers:

$x+ 3 - \frac{16}{x+3}$

$x- 3 - \frac{16}{x+3}$

$x- 3 + \frac{2}{x+3}$

$x+ 3 + \frac{16}{x+3}$

$x+ 3 + \frac{2}{x+3}$

Correct answer:

$x+ 3 - \frac{16}{x+3}$

Explanation:

Divide the leading coefficients to get the first term of the quotient:

$x^{2} \div x = x$ , the first term of the quotient

Multiply this term by the divisor, and subtract the product from the dividend:

$x (x+3) = x^{2} + 3x$

$\left (x^{2} + 6x -7 \right ) - (x^{2}+3x) = 3x -7$

Repeat these steps with the differences until the difference is an integer. As it turns out, we need to repeat only once:

$3x\div x = 3$ , the second term of the quotient

3 (x+3) = 3x + 9

(3x -7) - (3x + 9) = -16 , the remainder

Putting it all together, the quotient can be written as $x+ 3 - \frac{16}{x+3}$ .

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Example Question #1 : Dividing Polynomials

Simplify the following expression:

$\frac{(x-25)}{(x^2-625)}$

Possible Answers:

$\frac{1}{x-25}$

$\frac{1}{x^2+25}$

{x+25}

$\frac{1}{x+25}$

Correct answer:

$\frac{1}{x+25}$

Explanation:

Simplify the following expression:

$\frac{(x-25)}{(x^2-625)}$

To begin, we need to recognize the bottom as a difference of squares. Rewrite it as follows.

$\frac{(x-25)}{(x+25)(x-25)}=\frac{1}{x+25}$

So our answer is:

$\frac{1}{x+25}$

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Example Question #1 : Dividing Polynomials

Simplify the following expression:

$\frac{(3x)(3x^2+5x)}{x}$

Possible Answers:

9x^2+15x

9x+15

9x^3+15x^2

3x^2+5x

Correct answer:

9x^2+15x

Explanation:

Simplify the following expression:

$\frac{(3x)(3x^2+5x)}{x}$

First, let's multiply the 3x through:

$\frac{(3x)(3x^2+5x)}{x}=\frac{9x^3+15x^2}{x}$

Next, divide out the x from the bottom:

$\frac{9x^3+15x^2}{x}=9x^2+15x$

So our answer is:

9x^2+15x

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Example Question #2 : Dividing Polynomials

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{2}+30x+225}{x+15}\end{align*}$

Possible Answers:

x+5

x+3

x+15

x+23

Correct answer:

x+15

Explanation:

$\begin{align*}&\textbf{To perform a polynomial long division, it is often}\\&\textbf{helpful to space terms into columns ordered by power.}\\&\textbf{This includes zero terms. For example: }x^2+0x+5\\&\textbf{When performing each division step, choose a coefficient}\\&\textbf{Which eliminates the highest ordered term each time.}\end{align*}\begin{align*}&&&&&&&&\end{align*}\begin{align*}&&&&\mathbf{x}&&\mathbf{+15}\\x&&+15|x^{2}&&+30x&&+225\\&&-(x^{2}&&+15x)\\&&&&15x&&+225\\&&&&-(15x&&+225)\\\end{align*}$

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Example Question #7 : Polynomial Functions

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{2}+x-30}{x-5}\end{align*}$

Possible Answers:

x+1

x+6

x-6

x-26

Correct answer:

x+6

Explanation:

$\begin{align*}&\textbf{To perform a polynomial long division, it is often}\\&\textbf{helpful to space terms into columns ordered by power.}\\&\textbf{This includes zero terms. For example: }x^2+0x+5\\&\textbf{When performing each division step, choose a coefficient}\\&\textbf{Which eliminates the highest ordered term each time.}\end{align*}\begin{align*}&&&&&&&&\end{align*}\begin{align*}&&&&\mathbf{x}&&\mathbf{+6}\\x&&-5|x^{2}&&+x&&-30\\&&-(x^{2}&&-5x)\\&&&&6x&&-30\\&&&&-(6x&&-30)\\\end{align*}$

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Example Question #8 : Dividing Polynomials

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{2}+24x+143}{x+13}\end{align*}$

Possible Answers:

x-27

x-5

x+25

x+11

Correct answer:

x+11

Explanation:

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Example Question #181 : College Algebra

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{5}+x^{4}-353x^{3}-641x^{2}+24832x+98560}{x^{2}-9x-112}\end{align*}$

Possible Answers:

$x^{3}-25x^{2}-22x+27$

$x^{3}-7x^{2}-16x-6$

$x^{3}+10x^{2}-151x-880$

$x^{3}+28x^{2}+5x-27$

Correct answer:

$x^{3}+10x^{2}-151x-880$

Explanation:

$\begin{align*}&\textbf{To perform a polynomial}\\&\textbf{long division, it is often}\\&\textbf{helpful to space terms}\\&\textbf{into columns ordered by power.}\\&\textbf{This includes zero terms.}\\&\textbf{For example: }x^2+0x+5\\&\textbf{When performing each}\\&\textbf{division step, choose a coefficient}\\&\textbf{Which eliminates the highest}\\&\textbf{ordered term each time.}\end{align*}\begin{align*}&&&&&&&&\end{align*}\begin{align*}&&&&&&&&\mathbf{x^{3}}&&\mathbf{+10x^{2}}&&\mathbf{-151x}&&\mathbf{-880}\\x^{2}&&-9x&&-112|x^{5}&&+x^{4}&&-353x^{3}&&-641x^{2}&&+24832x&&+98560\\&&&&-(x^{5}&&-9x^{4}&&-112x^{3})\\&&&&&&10x^{4}&&-241x^{3}&&-641x^{2}\\&&&&&&-(10x^{4}&&-90x^{3}&&-1120x^{2})\\&&&&&&&&-151x^{3}&&479x^{2}&&24832x\\&&&&&&&&-(-151x^{3}&&1359x^{2}&&16912x)\\&&&&&&&&&&-880x^{2}&&7920x&&98560\\&&&&&&&&&&-(-880x^{2}&&7920x&&98560)\\\end{align*}$

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Example Question #182 : College Algebra

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{2}+7x-260}{x-13}\end{align*}$

Possible Answers:

x-20

x+9

x+20

x-28

Correct answer:

x+20

Explanation:

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Example Question #11 : Dividing Polynomials

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{4}-2x^{3}-84x^{2}+8x+320}{x^{2}-4}\end{align*}$

Possible Answers:

$x^{2}-19x-8$

$x^{2}-2x-80$

$x^{2}+8x+17$

$x^{2}-19x+11$

Correct answer:

$x^{2}-2x-80$

Explanation:

$\begin{align*}&\textbf{To perform a polynomial}\\&\textbf{long division, it is often}\\&\textbf{helpful to space terms}\\&\textbf{into columns ordered by power.}\\&\textbf{This includes zero terms.}\\&\textbf{For example: }x^2+0x+5\\&\textbf{When performing each}\\&\textbf{division step, choose a coefficient}\\&\textbf{Which eliminates the highest}\\&\textbf{ordered term each time.}\end{align*}\begin{align*}&&&&&&&&\end{align*}\begin{align*}&&&&&&&&\mathbf{x^{2}}&&\mathbf{-2x}&&\mathbf{-80}\\x^{2}&&+0&&-4|x^{4}&&-2x^{3}&&-84x^{2}&&+8x&&+320\\&&&&-(x^{4}&&+0&&-4x^{2})\\&&&&&&-2x^{3}&&+-80x^{2}&&8x\\&&&&&&-(-2x^{3}&&+0&&8x)\\&&&&&&&&-80x^{2}&&+0&&320\\&&&&&&&&-(-80x^{2}&&+0&&320)\\\end{align*}$

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Example Question #13 : Polynomial Functions

$\begin{align*}&\text{Simplify the expresssion:}\\&\frac{x^{2}-10x-11}{x-11}\end{align*}$

Possible Answers:

x-12

x-3

x+1

x-4

Correct answer:

x+1

Explanation:

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College Algebra : College Algebra

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #21 : Simplifying Polynomials

Example Question #1 : Dividing Polynomials

Example Question #1 : Dividing Polynomials

Example Question #2 : Dividing Polynomials

Example Question #7 : Polynomial Functions

Example Question #8 : Dividing Polynomials

Example Question #181 : College Algebra

Example Question #182 : College Algebra

Example Question #11 : Dividing Polynomials

Example Question #13 : Polynomial Functions

All College Algebra Resources

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