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Algebra II : Solving Rational Expressions

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Example Question #21 : Adding And Subtracting Rational Expressions

Simplify: $\frac{6x-1}{x}-\frac{3+x}{x^3}$

Possible Answers:

$\frac{6x^2-17x+3}{x^2}$

$\frac{5x-4}{x^3}$

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

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

$\frac{6x^2-17x+3}{x}$

Correct answer:

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

Explanation:

Rewrite the fractions with the least common denominator. The least common denominator is x^3 . We will need both fractions to have this denominator in order to add or subtract the numerator.

$\frac{(6x-1)(x^2)}{x(x^2)}-\frac{3+x}{x^3}$

Distribute the numerator and denominator of the first term.

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

Combine the numerators as one fraction. Be sure to enclose the second term in parentheses.

$\frac{6x^3-x^2-(3+x)}{x^3}$

Simplify the numerator.

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

There are no common factors.

The answer is: $\frac{6x^3-x^2-x-3}{x^3}$

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Example Question #22 : Adding And Subtracting Rational Expressions

Simplify: $\frac{5}{x}-\frac{2x}{x+7}$

Possible Answers:

$\frac{2x^2-5x-35}{x+7}$

$-\frac{2x^2-5x-35}{x^2+7x}$

$\frac{2x-5}{x-7}$

$-\frac{2x^2+5x-35}{x^2+7x}$

$-\frac{2x^2-5x-35}{x+7}$

Correct answer:

$-\frac{2x^2-5x-35}{x^2+7x}$

Explanation:

In order to add the numerators, we will need the least common denominator.

Multiply the denominators together.

x(x+7)=x^2+7x

Convert both fractions by multiplying both the top and bottom by what was multiplied to get the denominator. Rewrite the fractions and combine as one single fraction.

$\frac{5(x+7)}{x(x+7)}-\frac{2x(x)}{(x)(x+7)} = \frac{5x+35-2x^2}{x^2+7x}$

Re-order the terms.

$\frac{-2x^2+5x+35}{x^2+7x}$

Pull out a common factor of negative one on the numerator.

$\frac{-1(2x^2-5x-35)}{x^2+7x} = -\frac{2x^2-5x-35}{x^2+7x}$

The answer is: $-\frac{2x^2-5x-35}{x^2+7x}$

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Example Question #23 : Adding And Subtracting Rational Expressions

Subtract the following expressions: $\frac{x}{2x+1}-\frac{1}{2x}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

In order to subtract the fractions, multiply both denominators together in order to obtain the least common denominator.

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

Simplify the numerators.

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

Combine the numerators.

$=\frac{2x^2-(2x+1)}{(2x+1)(2x)}=\frac{2x^2-2x-1}{4x^2+2x}$

The answer is: $\frac{2x^2-2x-1}{4x^2+2x}$

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Example Question #24 : Adding And Subtracting Rational Expressions

Solve: $\frac{2}{x+3}+\frac{8}{x}$

Possible Answers:

$\frac{14x}{x^2+3x}$

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

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

$\frac{10x+24}{x^2+3x}$

$\frac{10}{2x+3}$

Correct answer:

$\frac{10x+24}{x^2+3x}$

Explanation:

To simplify this expression, we will need to multiply both denominators together to find the least common denominator.

x(x+3) =x^2+3x

Convert both fractions to the common denominator.

$\frac{2}{x+3}+\frac{8}{x}=\frac{2x}{x^2+3x}+\frac{8(x+3)}{x^2+3x}$

8(x+3)=8x+24

Combine the fractions.

$\frac{2x+8x+24}{x^2+3x} = \frac{10x+24}{x^2+3x}$

The answer is: $\frac{10x+24}{x^2+3x}$

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Example Question #21 : Adding And Subtracting Rational Expressions

$\frac{4}{x+2}+\frac{9x}{x-2}$

Possible Answers:

$\frac{9x^2+22x+8}{x^2-4}$

$\frac{x^2+22x-8}{x^2-4}$

$\frac{9x^2+22x-8}{x^2-4}$

$\frac{9x^2+22x-8}{x-4}$

$\frac{9x^2+22x-8}{x^2-2}$

Correct answer:

$\frac{9x^2+22x-8}{x^2-4}$

Explanation:

To add these rational expressions, first identify the common denominator.

In this case, it's the product of the two denominators:

(x+2)(x-2)=x^2-4 .

Then, multiply the numerators to offset them for their new denominator:

$\frac{4(x-2)}{(x+2)(x-2)}+\frac{9x(x+2)}{(x-2)(x+2)}$ .

Then combine the numerators to get

4x-8+9x^2+18x=9x^2+22x-8 .

Thus, your final answer is:

$\frac{9x^2+22x-8}{x^2-4}.$

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Example Question #26 : Adding And Subtracting Rational Expressions

$\frac{4}{x^2}+\frac{9}{x-1}$

Possible Answers:

$\frac{9x^2+4x-4}{x^2(x-1)}$

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

$\frac{9x^2+4x-4}{x^2}$

$\frac{9x^2+4x}{x^2(x-1)}$

$\frac{9x^2+4x-4}{(x-1)}$

Correct answer:

$\frac{9x^2+4x-4}{x^2(x-1)}$

Explanation:

To add these two expressions, first identify the common denominator.

In this case, it's the product of the two, which is

x^2(x-1) .

Then, multiply the numerators to offset them for the new denominator:

$\frac{4(x-1)}{x^2(x-1)}+\frac{9(x^2)}{x^2(x-1)}$ .

Now, combine the numerators to get your answer of:

$\frac{9x^2+4x-4}{x^2(x-1)}.$

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Example Question #27 : Adding And Subtracting Rational Expressions

$\frac{9}{x}+\frac{2}{x^2}-\frac{5}{x^3}$

Possible Answers:

$\frac{9x^2-2x-5}{x^3}$

$\frac{9x^2+2x-5}{x}$

$\frac{9x^2+2x-5}{x^3}$

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

$\frac{9x^2+2x-5}{x^6}$

Correct answer:

$\frac{9x^2+2x-5}{x^3}$

Explanation:

First, identify the common denominator. In this case, it's x^3 .

Offset the first two numerators to make up for the new denominator:

$\frac{9(x^2)}{x^3}+\frac{2(x)}{x^3}-\frac{5}{x^3}$ .

Now, combine the numerators to get:

9x^2+2x-5 .

Thus, your final answer is:

$\frac{9x^2+2x-5}{x^3}.$

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Example Question #28 : Adding And Subtracting Rational Expressions

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Find the least common denominator by multiplying the denominators together.

(1-x)(3+x) = (1)(3)+(1)(x)+(-x)(3)+(-x)(x)

Simplify the terms.

(1-x)(3+x)= -x^2-2x+3

Convert both fractions.

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

Simplify the numerator and denominator.

$\frac{6+2x}{(1-x)(3+x)}-\frac{2-2x}{(3+x)(1-x)}$

Combine to form one fraction. Brace the numerator of the second term since we are subtracting a quantity.

$\frac{6+2x-[2-2x]}{(1-x)(3+x)} =\frac{6+2x-2+2x}{(1-x)(3+x)} =\frac{4x+4}{(1-x)(3+x)}$

Replace the denominator with the simplified form.

$\frac{4x+4}{(1-x)(3+x)} =\frac{4x+4}{-x^2-2x+3}$

Pull out a common factor of negative one from the denominator.

$\frac{4x+4}{-x^2-2x+3} = \frac{4x+4}{-1(x^2+2x-3)}$

This allows us to pull the negative sign out in front of the fraction.

The answer is: $-\frac{4x+4}{x^2+2x-3}$

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Example Question #29 : Adding And Subtracting Rational Expressions

Add the fractions: $\frac{2}{x}+\frac{2}{5x^2+3}$

Possible Answers:

$\frac{10x+8}{5x^2+3}$

$\frac{12x^2+6}{5x^3+3x}$

$\frac{10x^2+2x+6}{5x^3+3x}$

$\frac{10x^2+8}{5x^3+3x}$

$\frac{10x^2+2x+8}{5x^3+3x}$

Correct answer:

$\frac{10x^2+2x+6}{5x^3+3x}$

Explanation:

Determine the least common denominator by multiplying both denominator together. Convert the fractions.

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

Combine the fractions as one fractions and simplify the numerator.

$\frac{2(5x^2+3)+2x}{x(5x^2+3)}= \frac{10x^2+6+2x}{x(5x^2+3)}= \frac{10x^2+2x+6}{5x^3+3x}$

The answer is: $\frac{10x^2+2x+6}{5x^3+3x}$

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Example Question #30 : Adding And Subtracting Rational Expressions

Solve: $\frac{6}{2+x} -\frac{11}{x}$

Possible Answers:

$\frac{17x-22}{x^2+2}$

$-\frac{17x+22}{x^2+2}$

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

$\frac{17x+22}{x^2+2}$

$-\frac{5x+22}{x^2+2}$

Correct answer:

$-\frac{5x+22}{x^2+2}$

Explanation:

In order to simplify this expression, we will need to find the least common denominator.

Multiply the two denominators together.

x(x+2) = x^2+2x

Convert both fractions to have the same common denominator.

$\frac{6}{2+x} -\frac{11}{x}=\frac{6x}{x^2+2}-\frac{11(2+x)}{x^2+2}$

Simplify the second term.

$\frac{11(2+x)}{x^2+2} = \frac{22+11x}{x^2+2}$

Combine the fractions to one fraction. Use a parentheses to brace (22+11x) term in the numerator since we are subtracting a quantity.

$\frac{6x-(22+11x)}{x^2+2} = \frac{6x-22-11x}{x^2+2}$

Combine like-terms.

$\frac{-5x-22}{x^2+2}$

Factor out a negative one from the numerator.

$\frac{-1(5x+22)}{x^2+2}$

This allows us to pull the negative sign out in front of the fraction.

The answer is: $-\frac{5x+22}{x^2+2}$

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Algebra II : Solving Rational Expressions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Adding And Subtracting Rational Expressions

Example Question #22 : Adding And Subtracting Rational Expressions

Example Question #23 : Adding And Subtracting Rational Expressions

Example Question #24 : Adding And Subtracting Rational Expressions

Example Question #21 : Adding And Subtracting Rational Expressions

Example Question #26 : Adding And Subtracting Rational Expressions

Example Question #27 : Adding And Subtracting Rational Expressions

Example Question #28 : Adding And Subtracting Rational Expressions

Example Question #29 : Adding And Subtracting Rational Expressions

Example Question #30 : Adding And Subtracting Rational Expressions

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