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8.1: Rational Expressions

Difficulty Level: At Grade Created by: CK-12
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Name: __________________

Rational Expressions

  1. \begin{align*}\frac{1}{x} + \frac{1}{y}\end{align*}
    1. Add
    2. Check your answer by plugging \begin{align*}x=2\end{align*} and \begin{align*}y=4\end{align*} into both my original expression, and your simplified expression. Do not use calculators or decimals.
  2. \begin{align*}\frac{1}{x} - \frac{1}{y} = \end{align*}
  3. \begin{align*}\left (\frac{1}{x} \right ) \left (\frac{1}{y}\right ) = \end{align*}
  4. \begin{align*}\frac{\frac{1}{x}}{\frac{1}{y}} = \end{align*}
  5. \begin{align*}\frac{x^2 + 2x + 1}{x^2 - 1}\end{align*}
    1. Simplify
    2. Check your answer by plugging \begin{align*}x=3\end{align*} into both my original expression, and your simplified expression. Do not use calculators or decimals.
    3. Are there any \begin{align*}x-\end{align*}values for which the new expression does not equal the old expression?
  6. \begin{align*}\frac{2}{x^2 - 9} - \frac{4}{x^2 + 2x - 15} = \end{align*}
  7. \begin{align*}\frac{4x^2 - 25 }{x^2 + 2x + 1} \times \frac{x^2 + 4x + 3}{2x^2 + x - 15}\end{align*}
    1. Multiply
    2. Check your answer by plugging \begin{align*}x=-2\end{align*} into both my original expression, and your simplified expression. (If they don’t come out the same, you did something wrong!)
    3. Are there any \begin{align*}x-\end{align*}values for which the new expression does not equal the old expression?

Name: __________________

Homework—Rational Expressions

  1. \begin{align*}\frac{x^2 - 6x + 5}{x^2 - 16} \times \frac{x^2 + 8x + 16}{x^2 - 7x + 10}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  2. \begin{align*}\frac{\frac{2x^2 + 7x + 3}{x^3 + 4x}} {\frac{x^2 + x - 6}{3x}}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  3. \begin{align*}\frac{1}{x - 1} - \frac{1}{x + 1}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
    4. Test your answer by choosing an \begin{align*}x\end{align*} value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?
  4. \begin{align*}\frac{x - 3}{x^2 + 9x + 20} - \frac{x - 4}{x^2 + 8x + 15}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  5. \begin{align*}\frac{x + 1}{4x^2 - 9} + \frac{4x}{6x^2 - 9x}\end{align*}
    1. Simplify
    2. Test your answer by choosing an \begin{align*}x\end{align*} value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?

Name: __________________

Rational Equations

1. Suppose I tell you that \begin{align*}\frac{x}{36} = \frac{15}{36}\end{align*}. What are all the values \begin{align*}x\end{align*} can take that make this statement true?

OK, that was easy, wasn’t it? So the moral of that story is: rational equations are easy to solve, if they have a common denominator. If they don’t, of course, you just get one!

2. Now suppose I tell you that \begin{align*}\frac{x}{18} = \frac{15}{36}\end{align*}. What are all the values \begin{align*}x\end{align*} can take that make this statement true?

Hey, \begin{align*}x\end{align*} came out being a fraction. Can he do that?

Umm, yeah.

OK, one more pretty easy one.

3. \begin{align*}\frac{x^2 + 2}{21} = \frac{9}{7}\end{align*}

Did you get only one answer? Then look again—this one has two!

Once you are that far, you’ve got the general idea—get a common denominator, and then set the numerators equal. So let’s really get into it now...

4. \begin{align*}\frac{x + 2}{x + 3} = \frac{x + 5}{x + 4}\end{align*}

5. \begin{align*}\frac{2x + 6}{x + 3} = \frac{x + 5}{2x + 7}\end{align*}

6. \begin{align*}\frac{x + 3}{2x - 3} = \frac{x - 5}{x - 4}\end{align*}

a. Solve. You should end up with two answers.

b. Check both answers by plugging them into the original equation.

Name: __________________

Homework: Rational Expressions and Equations

  1. \begin{align*}\frac{1}{x - 1} - \frac{1}{2x}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
    4. Test your answer by choosing an \begin{align*}x\end{align*} value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?
  2. \begin{align*}\frac{3x^2 + 5x - 8}{6x + 16} \times \frac{4x^3 + x}{3x^3 - 3x}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  3. \begin{align*}\frac{2}{4x^3 - x} + \frac{3}{2x^3 + 3x^2 + x}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  4. \begin{align*}\frac{\frac{x^2 + 9}{x^2 - 4}} {\frac{x^2 + 7x + 12}{x^2 + 2x - 8}}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
    4. Test your answer by choosing an \begin{align*}x\end{align*} value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?
  5. \begin{align*}\frac{x}{x^2 - 25} + <\mathrm{something}> = \frac{x^2 + 1}{x^3 - 9x^2 + 20x}\end{align*}
    1. What is the something?
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  6. \begin{align*}\frac{x - 6}{x - 3} = \frac{x + 18}{2x + 7}\end{align*}
    1. Solve for \begin{align*}x\end{align*}. You should get two answers.
    2. Check by plugging one of your answers back into the original equation.

Name: __________________

Dividing Polynomials

  1. \begin{align*}\frac{28k^3 p - 42 kp^2 + 56kp^3}{14 kp} = \end{align*}
  2. \begin{align*}\frac{x^2 - 12x - 45}{x + 3} =\end{align*}
    1. \begin{align*}\frac{2y^2 + y - 16}{y - 3} = \end{align*}
    2. Test your answer by choosing a number for \begin{align*}y\end{align*} and seeing if you get the same answer.
  3. \begin{align*}\frac{2h^3 - 5h^2 + 22h + 51}{2h + 3} = \end{align*}
  4. \begin{align*}\frac{2x^3 - 4x^2 + 6x - 15}{x^2 + 3} = \end{align*}
  5. \begin{align*}\frac{x^3 - 4x^2}{x - 4} =\end{align*}
    1. \begin{align*}\frac{x^3 - 27}{x - 3} = \end{align*}
    2. Test your answer by multiplying back.
  6. After dividing two polynomials, I get the answer \begin{align*}r^2 - 6r + 9- \frac{1}{r - 3}\end{align*}. What two polynomials did I divide?

Name: __________________

Sample Test: Rational Expressions

  1. \begin{align*}\frac{x - 3}{x^2 + 9x + 20} - \frac{x - 4}{x^2 + 8x + 15}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  2. \begin{align*}\frac{2}{x^2 - 1} + \frac{x}{x^2 - 2x + 1}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  3. \begin{align*}\frac{4x^3 - 9x}{x^2 - 3x - 10} \times \frac{2x^2 - 20x + 50}{6x^2 - 9x}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  4. \begin{align*}\frac{\frac{1}{x}}{\frac{x - 1}{x^2}}\end{align*}
    1. Simplify
    2. What values of \begin{align*}x\end{align*} are not allowed in the original expression?
    3. What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
  5. \begin{align*}\frac{x - 1}{2x - 1} = \frac{x+7}{7x + 4}\end{align*}
    1. Solve for \begin{align*}x\end{align*}.
    2. Test one of your answers and show that it works in the original expression. (No credit unless you show your work!)
  6. \begin{align*}\frac{6x^3 - 5x^2 - 5x + 34}{2x + 3}\end{align*}
    1. Solve by long division.
    2. Check your answer (show your work!!!).

Extra credit: I am thinking of two numbers, \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, that have this curious property: their sum is the same as their product. (Sum means “add them”; product means “multiply them.”)

a. Can you find any such pairs?

b. To generalize: if one of my numbers is \begin{align*}x\end{align*}, can you find a general formula that will always give me the other one?

c. Is there any number \begin{align*}x\end{align*} that has no possible \begin{align*}y\end{align*} to work with?

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Date Created:
Feb 23, 2012
Last Modified:
Apr 29, 2014
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