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College Algebra : Polynomial Functions

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Example Questions

Example Question #2 : Simplifying Polynomials

Divide the trinomial below by .

$18x^{2}+ 9x -3$

Possible Answers:

$2x^{2}+ x -\frac{1}{3}$

$2x+1-\frac{1}{3x}$

$-3x+1+\frac{1}{2x}$

$2x+1-\frac{3}{x}$

-x+1

Correct answer:

$2x+1-\frac{1}{3x}$

Explanation:

$18x^{2}+ 9x -3$

We can accomplish this division by re-writing the problem as a fraction.

$\frac{18x^2+9x-3}{9x}$

The denominator will distribute, allowing us to address each element separately.

$\frac{18x^{2}+ 9x -3}{9x}=\frac{18x^{2}}{9x}+\frac{ 9x}{9x}-\frac{3}{9x}$

Now we can cancel common factors to find our answer.

$(\frac{18}{9}* \frac{x^{2}}{x})+1-(\frac{3}{9}* \frac{1}{x})$

$2x+1-\frac{1}{3x}$

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Example Question #1 : Polynomial Functions

Simplify:

$y=\frac{(x^2-16)(x+7)}{x+4}+(x-1)^2+13$

Possible Answers:

$y=\frac{(x^2-16)(x+7)}{x+4}+x^2-2x+14$

y=x^2-x-14

y=2x^2+x-14

$y=\frac{x^3+7x^2-16x-112}{x+4}+(x-1)^2+13$

y=2(x^2+x-7)

Correct answer:

y=2x^2+x-14

Explanation:

$y=\frac{(x^2-16)(x+7)}{x+4}+(x-1)^2+13$

First, factor the numerator of the quotient term by recognizing the difference of squares:

$y=\frac{(x-4)(x+4)(x+7)}{x+4}+(x-1)^2+13$

Cancel out the common term from the numerator and denominator:

y=(x-4)(x+7)+(x-1)^2+13

FOIL (First Outer Inner Last) the first two terms of the equation:

y=x^2+7x-4x-28+x^2-2x+1+13

Combine like terms:

y=2x^2+x-14

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Example Question #2 : Dividing Polynomials

Divide:

$(x^{3} -23x+10) \div (x-5)$

Possible Answers:

$x^{2} + 5x + 2 + \frac{20}{x-5}$

$x^{2}+1+\frac{12}{x-5}$

$x^{2} + 5x + 2$

$x^{2} - 5x + 16 - \frac{73}{x-5}$

$x^{2} - 5x + 16 + \frac{87}{x-5}$

Correct answer:

$x^{2} + 5x + 2 + \frac{20}{x-5}$

Explanation:

First, rewrite this problem so that the missing $x^{2}$ term is replaced by $0x^{2}$

$(x^{3} + 0x^{2} -23x+10) \div (x-5)$

Divide the leading coefficients:

$x^{3} \div x = x^{2}$ , the first term of the quotient

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

$x^{2} (x-5) = x^{3} -5x^{2}$

$(x^{3} + 0x^{2} -23x+10) - (x^{3} -5x^{2}) = 5x^{2}-23x+10$

Repeat this process with each difference:

$5x^{2} \div x = 5x$ , the second term of the quotient

$5x (x-5) = 5 x^{2} -25$

$5x^{2}-23x+10 - ( 5x^{2} -25x) = 2x+10$

One more time:

$2x \div x = 2$ , the third term of the quotient

2 (x-5) = 2x-10

2x+10 - (2x-10) = 20 , the remainder

The quotient is $x^{2} + 5x + 2$ and the remainder is ; this can be rewritten as a quotient of

$x^{2} + 5x + 2 + \frac{20}{x-5}$

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Example Question #2 : Polynomial Functions

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 #3 : Dividing Polynomials

Simplify the following expression:

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

Possible Answers:

{x+25}

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

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

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

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 #4 : 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 #4 : Polynomial Functions

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

Possible Answers:

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

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

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

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

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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College Algebra : Polynomial Functions

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #2 : Simplifying Polynomials

Example Question #1 : Polynomial Functions

Example Question #2 : Dividing Polynomials

Example Question #2 : Polynomial Functions

Example Question #3 : Dividing Polynomials

Example Question #4 : Dividing Polynomials

Example Question #2 : Dividing Polynomials

Example Question #7 : Polynomial Functions

Example Question #8 : Dividing Polynomials

Example Question #4 : Polynomial Functions

All College Algebra Resources

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