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College Algebra : College Algebra

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Example Questions

Example Question #14 : Polynomial Functions

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

Possible Answers:

$x^{2}+23x+3$

$x^{2}-9x+27$

$x^{2}-x-30$

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

Correct answer:

$x^{2}-x-30$

Explanation:

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

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Example Question #15 : Polynomial Functions

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

Possible Answers:

$x^{2}-17x-4$

$x^{2}+21x-19$

$x^{2}-17x-20$

$x^{2}+12x+11$

Correct answer:

$x^{2}+12x+11$

Explanation:

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

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Example Question #16 : Polynomial Functions

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

Possible Answers:

x+29

x-24

x-4

x-17

Correct answer:

x-17

Explanation:

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

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Example Question #11 : Dividing Polynomials

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

Possible Answers:

$x^{2}+6x-40$

$x^{2}+13x-17$

$x^{2}-11x-5$

$x^{2}-23x-12$

Correct answer:

$x^{2}+6x-40$

Explanation:

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

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Example Question #18 : Polynomial Functions

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

Possible Answers:

$x^{3}-16x^{2}-3x+28$

$x^{3}+14x^{2}-x+5$

$x^{3}+3x^{2}+x-16$

$x^{3}-11x^{2}-354x+4104$

Correct answer:

$x^{3}-11x^{2}-354x+4104$

Explanation:

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

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Example Question #19 : Polynomial Functions

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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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Example Question #20 : Polynomial Functions

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

Possible Answers:

$x-4+\frac{6}{x+1}$

$x-4-\frac{6}{x+1}$

$-x-4-\frac{6}{x+1}$

$x+4-\frac{6}{x+1}$

Correct answer:

$x-4-\frac{6}{x+1}$

Explanation:

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

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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Example Question #21 : Polynomial Functions

Divide the polynomial: $\frac{x^2-4x-45}{x^2-12x+27}$

Possible Answers:

$x^2+\frac{1}{3}x -\frac{5}{3}$

$\frac{x-9}{x-3}$

$\frac{x+5}{x-3}$

$\frac{1}{3}x -\frac{5}{3}$

$\frac{x-5}{x+3}$

Correct answer:

$\frac{x+5}{x-3}$

Explanation:

To divide the polynomials, we will first need to factorize the top and bottom.

$\frac{x^2-4x-45}{x^2-12x+27} = \frac{(x-9)(x+5)}{(x-9)(x-3)}$

The answer is: $\frac{x+5}{x-3}$

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Example Question #21 : Polynomial Functions

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=- 297x - 270x^{2} - 27\\&\text{crosses the x-axis.}\end{align*}$

Possible Answers:

x=540,54

$x=-1,-\frac{1}{10}$

$x=1,\frac{1}{10}$

x=-540,-54

Correct answer:

$x=-1,-\frac{1}{10}$

Explanation:

$\begin{align*}&\text{The function }f(x)=- 297x - 270x^{2} - 27\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-27\cdot (x + 1)\cdot (10x + 1)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-297)\pm\sqrt{(-297)^2-4(-270)(-27)}}{2(-270)}=\frac{297\pm243}{-540}\\&\text{Either way, we find that the function crosses the x-axis at}\ x=-1\text{ and }-\frac{1}{10}.\end{align*}$

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Example Question #22 : Polynomial Functions

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=- 822x - 288x^{2} - 567\\&\text{has a zero value.}\end{align*}$

Possible Answers:

$x=-\frac{7}{6},-\frac{27}{16}$

x=-972,-672

$x=\frac{7}{6},\frac{27}{16}$

x=972,672

Correct answer:

$x=-\frac{7}{6},-\frac{27}{16}$

Explanation:

$\begin{align*}&\text{Given }f(x)=- 822x - 288x^{2} - 567\text{, to find when it has a value of}\\&\text{zero, we need a method to determine the x values that}\\&\text{make this happen. We can either factor the equation:}\\&-3\cdot (16x + 27)\cdot (6x + 7)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-822)\pm\sqrt{(-822)^2-4(-288)(-567)}}{2(-288)}=\frac{822\pm150}{-576}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=-\frac{7}{6}\text{ and }-\frac{27}{16}.\end{align*}$

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College Algebra : College Algebra

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #14 : Polynomial Functions

Example Question #15 : Polynomial Functions

Example Question #16 : Polynomial Functions

Example Question #11 : Dividing Polynomials

Example Question #18 : Polynomial Functions

Example Question #19 : Polynomial Functions

Example Question #20 : Polynomial Functions

Example Question #21 : Polynomial Functions

Example Question #21 : Polynomial Functions

Example Question #22 : Polynomial Functions

All College Algebra Resources

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