Polynomial Functions - College Algebra

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Question

Simplify:

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

Answer

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

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:

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

Combine like terms:

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Question

Divide the trinomial below by .

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

Answer

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

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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Question

Divide:

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

Answer

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

, the remainder

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

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Question

Divide:

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

Answer

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

, 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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Question

Simplify the following expression:

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

Answer

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:

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Question

Simplify the following expression:

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

Answer

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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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&\mathbf{x}&&\mathbf{+11}\x&&+13|x^{2}&&+24x&&+143\&&-(x^{2}&&+13x)\&&&&11x&&+143\&&&&-(11x&&+143)\\end{align}$

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Question

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

Answer

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

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Question

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

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&\mathbf{x}&&\mathbf{+20}\x&&-13|x^{2}&&+7x&&-260\&&-(x^{2}&&-13x)\&&&&20x&&-260\&&&&-(20x&&-260)\\end{align}$

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Question

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

Answer

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

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Question

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

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&\mathbf{x}&&\mathbf{+1}\x&&-11|x^{2}&&-10x&&-11\&&-(x^{2}&&-11x)\&&&&x&&-11\&&&&-(x&&-11)\\end{align}$

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Question

$\begin{align}&\text{Simplify the expresssion:}\&\frac{x^{4}-26x^{3}+151x^{2}+594x-4680}{x^{2}-25x+156}\end{align}$

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&&&&&\mathbf{x^{2}}&&\mathbf{-x}&&\mathbf{-30}\x^{2}&&-25x&&+156|x^{4}&&-26x^{3}&&+151x^{2}&&+594x&&-4680\&&&&-(x^{4}&&-25x^{3}&&+156x^{2})\&&&&&&-x^{3}&&-5x^{2}&&+594x\&&&&&&-(-x^{3}&&25x^{2}&&+-156x)\&&&&&&&&-30x^{2}&&750x&&+-4680\&&&&&&&&-(-30x^{2}&&750x&&+-4680)\\end{align}$

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Question

$\begin{align}&\text{Simplify the expresssion:}\&\frac{x^{4}+32x^{3}+347x^{2}+1372x+1056}{x^{2}+20x+96}\end{align}$

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&&&&&\mathbf{x^{2}}&&\mathbf{+12x}&&\mathbf{+11}\x^{2}&&+20x&&+96|x^{4}&&+32x^{3}&&+347x^{2}&&+1372x&&+1056\&&&&-(x^{4}&&+20x^{3}&&+96x^{2})\&&&&&&12x^{3}&&+251x^{2}&&+1372x\&&&&&&-(12x^{3}&&+240x^{2}&&+1152x)\&&&&&&&&11x^{2}&&+220x&&+1056\&&&&&&&&-(11x^{2}&&+220x&&+1056)\\end{align}$

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Question

$\begin{align}&\text{Simplify the expresssion:}\&\frac{x^{3}-x^{2}-233x-663}{x^{2}+16x+39}\end{align}$

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&&&&&\mathbf{x}&&\mathbf{-17}\x^{2}&&+16x&&+39|x^{3}&&-x^{2}&&-233x&&-663\&&&&-(x^{3}&&+16x^{2}&&+39x)\&&&&&&-17x^{2}&&+-272x&&+-663\&&&&&&-(-17x^{2}&&+-272x&&+-663)\\end{align}$

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Question

$\begin{align}&\text{Simplify the expresssion:}\&\frac{x^{3}+2x^{2}-64x+160}{x-4}\end{align}$

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&\mathbf{x^{2}}&&\mathbf{+6x}&&\mathbf{-40}\x&&-4|x^{3}&&+2x^{2}&&-64x&&+160\&&-(x^{3}&&-4x^{2})\&&&&6x^{2}&&-64x\&&&&-(6x^{2}&&-24x)\&&&&&&-40x&&160\&&&&&&-(-40x&&160)\\end{align}$

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Question

$\begin{align}&\text{Simplify the expresssion:}\&\frac{x^{4}-x^{3}-464x^{2}+564x+41040}{x+10}\end{align}$

Answer

$\begin{align}&\textbf{To perform a polynomial}\&\textbf{long division, it is often}\&\textbf{helpful to space terms}\&\textbf{into columns ordered by power.}\&\textbf{This includes zero terms.}\&\textbf{For example: }x^2+0x+5\&\textbf{When performing each}\&\textbf{division step, choose a coefficient}\&\textbf{Which eliminates the highest}\&\textbf{ordered term each time.}\end{align}\begin{align}&&&&&&&&\end{align}\begin{align}&&&&\mathbf{x^{3}}&&\mathbf{-11x^{2}}&&\mathbf{-354x}&&\mathbf{+4104}\x&&+10|x^{4}&&-x^{3}&&-464x^{2}&&+564x&&+41040\&&-(x^{4}&&+10x^{3})\&&&&-11x^{3}&&+-464x^{2}\&&&&-(-11x^{3}&&+-110x^{2})\&&&&&&-354x^{2}&&+564x\&&&&&&-(-354x^{2}&&+-3540x)\&&&&&&&&4104x&&+41040\&&&&&&&&-(4104x&&+41040)\\end{align}$

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Question

Perform the following operation: $3x^{2}-2x+1\div x-1$

Answer

Use synthetic division to for this polynomial

The initial set-up is shown below

$\right 1\rfloor$

First, bring down the leading coefficient:

$\right 1\rfloor$

$\underline{ }$

Multiply 1, by 3, and then add this to the second column:

$\right 1\rfloor$

$\underline{ }$

Next, multiply 1, by 1, and add this to the third column:

$\right 1\rfloor$

$\underline{ }$ ____

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

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Question

Simplify the following polynomial: $\frac{x^{2}-3x-10}{x+1}$

Answer

$\frac{x^{2}-3x-10}{x+1}$

Determine if there are any common factors between the numerator and the denominator:

$\frac{x^{2}-3x-10}{x+1}=\frac{(x-5)(x+2)}{x+1}$

There are no common factors, so we use synthetic division to simplify the polynomial:

$\right-1 \rfloor$

_________

Bring down the 1, from the first column:

$\right-1 \rfloor$

_

Multiply 1 by -1, and add the product to -3:

$\right-1 \rfloor$

_

Multiply -4 by -1, and add the product to -10:

$\right-1 \rfloor$

_________

Solution:

$\frac{x^{2}-3x-10}{x+1}=$ $x-4-\frac{6}{x+1}$

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