College Algebra : Polynomial Functions

Study concepts, example questions & explanations for College Algebra

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Example Questions

Example Question #3 : Polynomials

Divide the trinomial below by \displaystyle 9x.

\displaystyle 18x^{2}+ 9x -3

Possible Answers:

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

\displaystyle -x+1

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

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

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

Correct answer:

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

Explanation:

\displaystyle 18x^{2}+ 9x -3

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

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

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

\displaystyle \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.

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

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

Example Question #1 : Polynomial Functions

Simplify:

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

Possible Answers:

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

\displaystyle y=2x^2+x-14

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

\displaystyle y=x^2-x-14

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

Correct answer:

\displaystyle y=2x^2+x-14

Explanation:

\displaystyle 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:

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

Cancel out the common term from the numerator and denominator:

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

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

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

Combine like terms:

\displaystyle y=2x^2+x-14

Example Question #1 : Polynomial Functions

Divide:

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

Possible Answers:

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

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

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

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

\displaystyle x^{2} + 5x + 2

Correct answer:

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

Explanation:

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

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

Divide the leading coefficients:

\displaystyle 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:

\displaystyle x^{2} (x-5) = x^{3} -5x^{2}

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

Repeat this process with each difference:

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

\displaystyle 5x (x-5) = 5 x^{2} -25

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

One more time:

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

\displaystyle 2 (x-5) = 2x-10

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

The quotient is \displaystyle x^{2} + 5x + 2 and the remainder is \displaystyle 20; this can be rewritten as a quotient of 

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

Example Question #2 : Polynomial Functions

Divide:

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

 

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

\displaystyle x (x+3) = x^{2} + 3x

\displaystyle \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:

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

\displaystyle 3 (x+3) = 3x + 9

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

 

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

Example Question #1 : Polynomial Functions

Simplify the following expression:

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

Possible Answers:

\displaystyle {x+25}

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

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

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

Correct answer:

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

Explanation:

Simplify the following expression:

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

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

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

So our answer is:

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

Example Question #5 : Polynomial Functions

Simplify the following expression:

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

Possible Answers:

\displaystyle 3x^2+5x

\displaystyle 9x^2+15x

\displaystyle 9x+15

\displaystyle 9x^3+15x^2

Correct answer:

\displaystyle 9x^2+15x

Explanation:

Simplify the following expression:

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

First, let's multiply the 3x through:

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

Next, divide out the x from the bottom:

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

So our answer is:

\displaystyle 9x^2+15x

Example Question #3 : Polynomial Functions

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

Possible Answers:

\displaystyle x+5

\displaystyle x+15

\displaystyle x+3

\displaystyle x+23

Correct answer:

\displaystyle x+15

Explanation:

\displaystyle \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*}

Example Question #4 : Polynomial Functions

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

Possible Answers:

\displaystyle x+1

\displaystyle x-26

\displaystyle x-6

\displaystyle x+6

Correct answer:

\displaystyle x+6

Explanation:

\displaystyle \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*}

Example Question #5 : Polynomial Functions

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

Possible Answers:

\displaystyle x+25

\displaystyle x-27

\displaystyle x-5

\displaystyle x+11

Correct answer:

\displaystyle x+11

Explanation:

\displaystyle \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*}

Example Question #181 : College Algebra

\displaystyle \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:

\displaystyle x^{3}+10x^{2}-151x-880

\displaystyle x^{3}+28x^{2}+5x-27

\displaystyle x^{3}-7x^{2}-16x-6

\displaystyle x^{3}-25x^{2}-22x+27

Correct answer:

\displaystyle x^{3}+10x^{2}-151x-880

Explanation:

\displaystyle \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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