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College Algebra : Rational Expressions

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Example Question #101 : Review And Other Topics

Simplify the following:

$\frac{6x^{2}-x-1}{12x^{2}+x-1}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

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

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 (3x+1)

Simplified, the equation is:

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

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Example Question #12 : Rational Expressions

Simplify the following:

$\frac{16x^{2}-1}{4x^{2}-15x-4}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

The first step is to factor both polynomials

$\frac{16x^{2}-1}{4x^{2}-15x-4}$ becomes $\frac{(4x+1)(4x-1)}{(4x+1)(x-4)}$

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

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Example Question #11 : Rational Expressions

Simplify the following:

$\frac{2x^{2}+13x-7}{2x^{2}-17x+8}$

Possible Answers:

$\frac{(x+7)}{(x+8)}$

$\frac{(x-7)}{(x-8)}$

$\frac{(x+7)}{(x-8)}$

$\frac{(x+8)}{(x-7)}$

$\frac{(x-7)}{(x+8)}$

Correct answer:

$\frac{(x+7)}{(x-8)}$

Explanation:

The first step is to factor both polynomials

$\frac{2x^{2}+13x-7}{2x^{2}-17x+8}$ becomes $\frac{(2x-1)(x+7)}{(2x-1)(x-8)}$

Simplified, the equation is:

$\frac{(x+7)}{(x-8)}$

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Example Question #11 : Rational Expressions

Simplify the given expression:

$\frac{x^2+4x-12}{2x+12}$

Possible Answers:

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

x-2

$\frac{1}{x+6}$

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

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

Correct answer:

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

Explanation:

First we want to factor both the numerator and denominator:

$\frac{x^2+4x-12}{2x+12}$

For the numerator, we can use the quadratic formula:

$x=\frac{-(4)\pm \sqrt{4^{2}-4(1)(-12)}}{2(1)}$

$\\x=\frac{-4\pm \sqrt{16+48}}{2}\\x=\frac{-4\pm\sqrt{64}}{2}\\x=\frac{-4\pm8}{2}$

$\\x=\frac{-12}{2}\;or\;x=\frac{4}{2}\\\\x=-6\;or\;x=2$

For the denominator, we only need to factor out the , leaving us with the given factored expression:

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

Then we cancel out like terms, giving us a simplified expression:

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

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Example Question #381 : College Algebra

Simplify: $\frac{3x^5}{2x^5+x}$

Possible Answers:

$\frac{3}{2}$

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

$\textup{The rational expression cannot be factored.}$

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

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

Correct answer:

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

Explanation:

Factor an in the denominator.

$\frac{3x^5}{2x^5+x} = \frac{3x^5}{x(2x^4+1)}$

Cancel the common in the numerator and denominator.

The answer is: $\frac{3x^4}{2x^4+1}$

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Example Question #11 : Rational Expressions

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To add the terms of the numerator, we will need to have the same denominator.

Multiply both denominators together.

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

Multiply the numerators with what was multiplied on the denominator.

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

Combine like terms.

The answer is: $\frac{3x-1}{x^2-x}$

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Example Question #11 : Rational Expressions

Add/Subtract: $\frac {4}{5}-\frac {1}{4}$ $\frac {4}{5}-\frac {1}{4}+1$

Possible Answers:

$1\frac {11}{20}$

None of the Above

Correct answer:

$1\frac {11}{20}$

Explanation:

Step 1: Find the Least Common denominator of all the fractions..

LCD= 4*5*1=20

Step 2: Convert all fractions to denominator .

$\frac {4}{5}\equiv \frac {16}{20}$

$\frac {1}{4}\equiv \frac {5}{20}$

$1=\frac {20}{20}$

Step 3: Re-write all fractions in terms of the original equation...

$\frac {16}{20}-\frac {5}{20}+\frac {20}{20}$

Answer: $\frac {31}{20}$ or $1\frac {11}{20}$

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College Algebra : Rational Expressions

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #101 : Review And Other Topics

Example Question #12 : Rational Expressions

Example Question #11 : Rational Expressions

Example Question #11 : Rational Expressions

Example Question #381 : College Algebra

Example Question #11 : Rational Expressions

Example Question #11 : Rational Expressions

All College Algebra Resources

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