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Algebra 1 : How to divide polynomials

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Example Questions

Example Question #1 : Polynomial Operations

Simplify $\frac{x^{5}y^{10}z^{3}}{x^{8}y^{-6}z^{2}}$

Possible Answers:

$x^{13}y^{16}z^{5}$

$\frac{y^{16}z}{x^{3}}$

$x^{3}y^{4}z^{5}$

$\frac{x^{3}z}{y^{4}}$

$x^{3}y^{4}z$

Correct answer:

$\frac{y^{16}z}{x^{3}}$

Explanation:

When dividing exponents, you subtract exponents that share the same base, so

5-8=-3 and 10-(-6)=16 and 3-2=1 .

Do not forget to "add the opposite" when subtracting negative numbers).

Now, you have

$x^{-3}y^{16}z^{1}$

But you are not done yet! Remember, you do not want to have a negative exponent, and the way to turn the negative exponent into a positive exponent is to take its reciprocal, like this:

$x^{-3} = \frac{1}{x^{3}}$

You keep the rest of the equation in the numerator, leaving you with

$\frac{y^{16}z}{x^{3}}$

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Example Question #1 : How To Divide Polynomials

Simplify the rational expression.

$\frac{r^2s^9x^5}{r^4s^3x}$

Possible Answers:

$r^6s^{12}x^6$

s^3x^4

$r^{2}s^6x^4$

$r^{-2}s^3x^4$

$\frac{s^6x^4}{r^2}$

Correct answer:

$\frac{s^6x^4}{r^2}$

Explanation:

$\frac{r^2s^9x^5}{r^4s^3x}$

To simplify, we must use exponent rules. For exponents in fractions, we can subtract the exponent of the denominator from the exponent in the numerator.

$\frac{a^b}{a^c}=a^{b-c}$

With this rule, we can rewrite the problem.

$r^{2-4}s^{9-3}x^{5-1}$

$r^{-2}s^6x^4$

Remember that negative exponents get moved back to the denominator, turning them positive.

$\frac{s^6x^4}{r^2}$

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Example Question #3 : How To Divide Polynomials

Find the Greatest Common Factor (GCF) of the following polynomial:

$24x^{4}y^{3}-12x^{3}y^{4}+8x^{5}y^{2}-4x^{2}y$

Possible Answers:

$4x^{2}y^{2}$

$4x^{3}$

$4x^{3}y$

$4xy^{2}$

$4x^{2}y$

Correct answer:

$4x^{2}y$

Explanation:

4 goes into 24, 12, 8, and 4.

Similarly, the smallest exponent of x in the four terms is 2, and the smallest exponent of y in the four terms is 1.

Hence the GCF must be $4x^{2}y$ .

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Example Question #3 : How To Divide Polynomials

Divide:

$\left ( 4x^{2}y^{2}-8xy^{3}+16xy^{4} \right )\div \left ( 4x^{2}y \right )$

Possible Answers:

$y - \frac{2y^{2}}{x^{2}}+\frac{4y^{2}}{x^{2}}$

$y-\frac{2y^{2}}{x}+\frac{4y^{3}}{x}$

$y-\frac{2y}{x^{2}}+4\frac{y}{x^{3}}$

$1 - \frac{2y}{x} + \frac{4y^{3}}{x^{2}}$

$1-\frac{2y}{x}+\frac{4y^{2}}{x^{2}}$

Correct answer:

$y-\frac{2y^{2}}{x}+\frac{4y^{3}}{x}$

Explanation:

Divide each of the terms in the numerator by the denominator:

$\frac{4x^{2}y^{2}}{4x^{2}y}-\frac{8xy^{3}}{4x^{2}y} + \frac{16y^{4}}{4x^{2}y}$

Simplify each term above to get the final:

$y-\frac{2y^{2}}{x}+\frac{4y^{3}}{x}$

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Example Question #1 : Polynomial Operations

Find the quotient:

$\frac{2x^{3}+x^{2}-12x+9}{x+3}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The numerator can be factored into

$\left ( x-1 \right )\left ( x+3 \right )\left ( 2x-3 \right )$ ,

which when divided by $\left ( x+3 \right )$ ,

gives us $\left ( x-1 \right )\left ( 2x-3 \right ) = 2x^{2}-5x+3$ .

Alternate method: Long division of the numerator by the denominator gives the same answer.

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Example Question #4 : How To Divide Polynomials

Find the remainder:

$\frac{2x^{3}+x^{2}-12x+3}{2x-3}$

Possible Answers:

-6

$\frac{-6}{2x-3}$

$\frac{-6x}{2x-3}$

$\frac{6}{2x-3}$

$\frac{-6x^{2}}{2x-3}$

Correct answer:

$\frac{-6}{2x-3}$

Explanation:

When we divide a polynomial by another polynomial we get:

Quotient
Remainder (if one exists)

In our problem the long division results in:

A quotient of $x^{2}+2x-3$
A remainder of $\frac{-6}{2x-3}$

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Example Question #3 : How To Divide Polynomials

Divide:

$\frac{8x^{3}-1}{2x-1}$

Possible Answers:

$4x^{2}-2x+1$

$4x^{2}-2x-1$

$4x^{2}+2x+1$

$4x^{2} +1$

8x-1

Correct answer:

$4x^{2}+2x+1$

Explanation:

This can easily be solved by factoring using the difference of cubes formula:

$\left ( x^{3}-y^{3} \right ) = \left ( x-y \right )\left ( x^{2}+xy+y^{2} \right )$

First, convert the given polynomial into a difference of two cubes:

$\left ( \left ( 2x \right )^{3} -\left ( 1 \right )^{3}\right )$

Compare this with the difference of cubes formula above to get:

$\left ( 2x-1 \right )\left ( 4x^{2}+2x+1 \right )$

By dividing the above numerator by the given denominator we get:

$4x^{2}+2x+1$

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Example Question #2 : Polynomial Operations

Divide: $(5t^{3} - 40t^{2}+ 30t- 10) \div 10t$

Possible Answers:

$\frac{1}{2} t^{2}- 4t+ 3- \frac{1}{2t}$

$2 t^{2}- 4t+ 3- \frac{1}{t}$

$2 t^{2}- 5t+ 3$

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

$\frac{1}{2} t^{2}- 4t+ 3- \frac{1}{t}$

Correct answer:

$\frac{1}{2} t^{2}- 4t+ 3- \frac{1}{t}$

Explanation:

$(5t^{3} - 40t^{2}+ 30t- 10) \div 10t$

$= \frac{5t^{3}}{10t} - \frac{40t^{2}}{10t}+ \frac{30t}{10t}- \frac{10}{10t}$

Cancel:

$= \frac{t^{3-1}}{2} - \frac{4t^{2-1}}{1}+ \frac{3}{1}- \frac{1}{t}$

$= \frac{1}{2} t^{2}- 4t+ 3- \frac{1}{t}$

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Example Question #4 : How To Divide Polynomials

Divide: $\left ( 9x ^{2} + 21x - 18\right ) \div \left (-3x \right )$

Possible Answers:

$-3x -7+\frac{6}{x }$

$-3x -7-\frac{6}{x }$

$-3x +7-\frac{6}{x }$

3x -7

-9x -7

Correct answer:

$-3x -7+\frac{6}{x }$

Explanation:

$\left ( 9x ^{2} + 21x - 18\right ) \div \left (-3x \right )$

$=\frac{9x ^{2}}{-3x } +\frac{21x }{-3x }- \frac{18}{-3x }$

$=-\frac{9x ^{2}}{3x }-\frac{21x }{3x }+ \frac{18}{3x }$

Cancel:

$=-\frac{3x ^{2-1}}{1 }-\frac{7 }{1 }+ \frac{6}{x }$

$=-3x -7+\frac{6}{x }$

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Example Question #6 : How To Divide Polynomials

Simplify:

$\frac {21b^2 + 7b^3 - 14b}{7b}$

Possible Answers:

3b^2 + b^3 - 2

3b^3 + b^4 + 2b^2

3b^3 + b^4 - 2b^2

$b^{2}+3b-2$

3b + b^2 + 2

Correct answer:

$b^{2}+3b-2$

Explanation:

7 in the denominator is a common factor of the three coefficients in the numerator, which allows you to divide out the 7 from the denominator:

$\frac {3b^2 + b^3 - 2b}{b}$

Then divide by :

$={3b + b^2 - 2}$

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Algebra 1 : How to divide polynomials

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : Polynomial Operations

Example Question #1 : How To Divide Polynomials

Example Question #3 : How To Divide Polynomials

Example Question #3 : How To Divide Polynomials

Example Question #1 : Polynomial Operations

Example Question #4 : How To Divide Polynomials

Example Question #3 : How To Divide Polynomials

Example Question #2 : Polynomial Operations

Example Question #4 : How To Divide Polynomials

Example Question #6 : How To Divide Polynomials

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