College Algebra : Rational Expressions

Study concepts, example questions & explanations for College Algebra

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Example Questions

Example Question #1 : Rational Expressions

Rationalize the following fraction:

\displaystyle \frac{15}{\sqrt{5}}

Possible Answers:

\displaystyle 3\sqrt{5}

\displaystyle 15\sqrt{5}

\displaystyle 5\sqrt{3}

\displaystyle \frac{3\sqrt{5}}{5}

Correct answer:

\displaystyle 3\sqrt{5}

Explanation:

Rationalize the following fraction:

\displaystyle \frac{15}{\sqrt{5}}

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

\displaystyle \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

Example Question #92 : Review And Other Topics

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

First we need to factor both polynomials.

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

Becomes

 

\displaystyle \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 \displaystyle (x+3), removing that from the equation gives us

 

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

 

Example Question #93 : Review And Other Topics

Simplify the following

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials:

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

Once we remove that, we are left with:

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

 

Example Question #376 : College Algebra

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

\displaystyle \frac{x^{2}-2x-8}{x^{2}-x-12}   becomes \displaystyle \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 \displaystyle (x-4)

 

Simplified, the equation is:

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

Example Question #2 : Rational Expressions

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

 

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

Simplified, the equation is:

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

Example Question #2 : Rational Expressions

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

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

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 \displaystyle (4x+3)

Simplified, the equation is:

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

Example Question #382 : College Algebra

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

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


Simplified, the equation is:

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

Example Question #383 : College Algebra

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

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


Simplified, the equation is:

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

Example Question #384 : College Algebra

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

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


Simplified, the equation is:

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

Example Question #385 : College Algebra

Simplify the following:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

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

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 \displaystyle (5x+1)


Simplified, the equation is:

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

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