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Algebra 1 : Algebra 1

Study concepts, example questions & explanations for Algebra 1

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Example Questions

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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Example Question #11 : How To Divide Polynomials

Simplify:

$\frac{\left ( 6x^{2}\right ) ^{3} }{\left ( 6x^{3}\right ) ^{2}}$

Possible Answers:

$6x^{2}$

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

$\frac{1}{6x^{2}}$

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

Correct answer:

Explanation:

Use the properties of powers:

$\frac{\left ( 6x^{2}\right ) ^{3} }{\left ( 6x^{3}\right ) ^{2}} = \frac{ 6^{3}\left ( x^{2}\right ) ^{3} }{6^{2}\left ( x^{3}\right ) ^{2}} =\frac{ 216\cdot x^{2 \cdot 3} }{36\cdot x^{3 \cdot 2}} = \frac{ 216 x^{6} }{36 x^{6}}$

We can cancel now:

$\frac{ 216 x^{6} }{36 x^{6}} = \frac{ 216 \div 36 }{36 \div 36} = \frac{ 6 }{1} = 6$

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Example Question #11 : How To Divide Polynomials

$\frac{4x^{2}+6x+2}{2x+2}=$

Possible Answers:

x+4

2x+1

x+2

2x+6

2x+2

Correct answer:

2x+1

Explanation:

To easily divide the polynomials, factor the numerator to get

$\frac{(2x+2)(2x+1)}{2x+2}$

Then, you can cancel out 2x+2 since it is present in both the numerator and the denominator, leaving you with just 2x+1 .

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Example Question #13 : Polynomial Operations

$\frac{3x^{2}+7x+2}{x+2}$

Possible Answers:

x+1

3x+1

x+3

3x+5

3x+3

Correct answer:

3x+1

Explanation:

First, factor the numerator to get

$\frac{(3x+1)(x+2)}{x+2}$

Since x+2 is in both the numerator and the denominator, they cancel each other out. This leaves just 3x+1 as the simplified version of the expression.

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Example Question #12 : How To Divide Polynomials

Simplify this expression to its lowest terms: $\frac{15x^5y^4z^6}{3x^6y^3z^6}$

Possible Answers:

$\frac{5yz}{x}$

$\frac{5y}{xz}$

$\frac{5y}{x}$

5xz

Correct answer:

$\frac{5y}{x}$

Explanation:

When we divide a complex set of variables which are all being multiplied by another similar set of variables which are being multiplied (as in our problem!) we are able to handle each separate variable as it's own division problem. So, that the complex fraction

$\frac{15x^5y^4z^6}{3x^6y^3z^6}$

is really 4 separate fractions:

$\frac{15}{3}\cdot\frac{x^5}{x^6}\cdot\frac{y^4}{y^3}\cdot\frac{z^6}{z^6}$

And, when we divide like terms with exponents, we subtract the powers! So, we are able to eliminate the , we leave one in the denominator and one in the numerator! $\frac{15}{3}$ is 5, so we leave the 5 in the numerator as well. This leaves us with the answer:

$\frac{5y}{x}$

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Example Question #15 : Polynomial Operations

Simplify the fraction to its simplest terms: $\frac{24x^2y^3z^2}{48x^4y^4z^5}$

Possible Answers:

$\frac{1}{ x^2 z^3}$

$\frac{1}{ x^2 y z^3}$

$\frac{x}{2 y z^3}$

2 x^2 y z^3}

$\frac{1}{2 x^2 y z^3}$

Correct answer:

$\frac{1}{2 x^2 y z^3}$

Explanation:

When we divide a complex set of variables which are all being multiplied, by another similar set of variables which are being multiplied (as in our problem!) we are able to handle each separate variable as it's own division problem. So the complex fraction:

$\frac{24x^2y^3z^2}{48x^4y^4z^5}$

is really 4 separate fractions:

$\frac{24}{48}\cdot\frac{x^2}{x^4}\cdot\frac{y^3}{y^4}\cdot\frac{z^2}{z^5}$

And, when we divide like terms with exponents, we subtract the powers! So, we are able to do this and we leave x^2 , & z^3 in the denominator and, since there is nothing left in the numerator we put the placeholder, 1, there!

$\frac{24}{48}=\frac{1}{2}$ so we leave the 1 in the numerator and put the 2 as the coefficient in the denominator. Leaving us with the answer: $\frac{1}{2 x^2 y z^3}$

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Example Question #1 : Simplifying Polynomials

Divide the trinomial below by .

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

Possible Answers:

-x+1

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

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

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

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

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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Algebra 1 : Algebra 1

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

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

Example Question #11 : How To Divide Polynomials

Example Question #11 : How To Divide Polynomials

Example Question #13 : Polynomial Operations

Example Question #12 : How To Divide Polynomials

Example Question #15 : Polynomial Operations

Example Question #1 : Simplifying Polynomials

All Algebra 1 Resources

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