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College Algebra : Rational Expressions

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Example Questions

Example Question #1 : Rational Expressions

Rationalize the following fraction:

$\frac{15}{\sqrt{5}}$

Possible Answers:

$\frac{3\sqrt{5}}{5}$

$5\sqrt{3}$

$3\sqrt{5}$

$15\sqrt{5}$

Correct answer:

$3\sqrt{5}$

Explanation:

Rationalize the following fraction:

$\frac{15}{\sqrt{5}}$

To rationalize a denominator, we will multiply the top and bottom of the fraction by the denominator.

$\frac{15}{\sqrt{5}}*\frac{\sqrt{5}}{\sqrt{5}}=\frac{15\sqrt{5}}{(\sqrt{5})^2}=\frac{15\sqrt{5}}{5}=3\sqrt{5}$

And we have our answer

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Example Question #92 : Review And Other Topics

Simplify the following:

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

Possible Answers:

$\frac{x+3}{x+4}$

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

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

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

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

Correct answer:

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

Explanation:

First we need to factor both polynomials.

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

Becomes

$\frac{(x+3)(x-3)}{(x+3)(x-4)}$

Now we cancel out any variables that are in BOTH the numerator and denominator. Remember that if a group of variables/numbers are inside parenthesis, they are considered a single term.

The common term in this case is (x+3) , removing that from the equation gives us

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

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Example Question #93 : Review And Other Topics

Simplify the following

$\frac{2b^{2}+7b+5}{b^{2}-1}$

Possible Answers:

$\frac{2b-5}{b+1}$

$\frac{2b-5}{b-1}$

$\frac{2b+5}{b+1}$

$\frac{2b+5}{b-1}$

$\frac{b+5}{b-1}$

Correct answer:

$\frac{2b+5}{b-1}$

Explanation:

The first step is to factor both polynomials:

$\frac{2b^{2}+7b+5}{b^{2}-1}$ Becomes $\frac{(2b+5)(b+1)}{(b+1)(b-1)}$

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is (b+1)

Once we remove that, we are left with:

$\frac{2b+5}{b-1}$

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Example Question #2 : Rational Expressions

Simplify the following:

$\frac{x^{2}-2x-8}{x^{2}-x-12}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

$\frac{x^{2}-2x-8}{x^{2}-x-12}$ becomes $\frac{(x-4)(x+2)}{(x-4)(x+3)}$

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is (x-4)

Simplified, the equation is:

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

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Example Question #3 : Rational Expressions

Simplify the following:

$\frac{3x^{2}+11x+10}{3x^{2}-x-10}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

$\frac{3x^{2}+11x+10}{3x^{2}-x-10}$ becomes $\frac{(3x+5)(x+2)}{(3x+5)(x-2)}$

Now we cancel out any terms that are in both the numerator and denominator. Remember that any variables/numbers that are in parenthesis are considered a single term.

In this case, the common term is (3x+5)

Simplified, the equation is:

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

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Example Question #3 : Rational Expressions

Simplify the following:

$\frac{16x^{2}-9}{4x^{2}-17x-15}$

Possible Answers:

$\frac{(4x+3)}{(x-5)}$

$\frac{(4x-3)}{(x-5)}$

$\frac{(4x+3)}{(x+5)}$

$\frac{(x-5)}{(4x-3)}$

$\frac{(4x-3)}{(x+5)}$

Correct answer:

$\frac{(4x-3)}{(x-5)}$

Explanation:

The first step is to factor both polynomials

$\frac{16x^{2}-9}{4x^{2}-17x-15}$ becomes $\frac{(4x-3)(4x+3)}{(4x+3)(x-5)}$

Simplified, the equation is:

$\frac{(4x-3)}{(x-5)}$

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Example Question #382 : College Algebra

Simplify the following:

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

Possible Answers:

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

$\frac{(x-1)}{(x-4)}$

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

$\frac{(x+1)}{(x+4)}$

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

Correct answer:

$\frac{(x+1)}{(x+4)}$

Explanation:

The first step is to factor both polynomials

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

$\frac{(x+1)}{(x+4)}$

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Example Question #4 : Rational Expressions

Simplify the following:

$\frac{12x^{2}+11x+2}{6x^{2}+x-2}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

$\frac{12x^{2}+11x+2}{6x^{2}+x-2}$ becomes $\frac{(3x+2)(4x+1)}{(2x-1)(3x+2)}$

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

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Example Question #384 : College Algebra

Simplify the following:

$\frac{2x^{2}+x-15}{3x^{2}+4x-15}$

Possible Answers:

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

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

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

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

$\frac{(2x-5)}{(3x-5)}$

Correct answer:

$\frac{(2x-5)}{(3x-5)}$

Explanation:

The first step is to factor both polynomials

$\frac{2x^{2}+x-15}{3x^{2}+4x-15}$ becomes $\frac{(2x-5)(x+3)}{(3x-5)(x+3)}$

$\frac{(2x-5)}{(3x-5)}$

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Example Question #385 : College Algebra

Simplify the following:

$\frac{5x^{2}+31x+6}{5x^{2}-29x-6}$

Possible Answers:

$\frac{(x+6)}{(x-6)}$

$\frac{(5x+6)}{(3x-6)}$

$\frac{(5x+6)}{(x-6)}$

$\frac{(x-6)}{(x+6)}$

$\frac{(x+6)}{(5x-6)}$

Correct answer:

$\frac{(x+6)}{(x-6)}$

Explanation:

The first step is to factor both polynomials

$\frac{5x^{2}+31x+6}{5x^{2}-29x-6}$ becomes $\frac{(5x+1)(x+6)}{(5x+1)(x-6)}$

$\frac{(x+6)}{(x-6)}$

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College Algebra : Rational Expressions

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #1 : Rational Expressions

Example Question #92 : Review And Other Topics

Example Question #93 : Review And Other Topics

Example Question #2 : Rational Expressions

Example Question #3 : Rational Expressions

Example Question #3 : Rational Expressions

Example Question #382 : College Algebra

Example Question #4 : Rational Expressions

Example Question #384 : College Algebra

Example Question #385 : College Algebra

All College Algebra Resources

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