8.1: Rational Expressions
Name: __________________
Rational Expressions

\begin{align*}\frac{1}{x} + \frac{1}{y}\end{align*}
1x+1y  Add
 Check your answer by plugging \begin{align*}x=2\end{align*}
x=2 and \begin{align*}y=4\end{align*}y=4 into both my original expression, and your simplified expression. Do not use calculators or decimals.

\begin{align*}\frac{1}{x}  \frac{1}{y} = \end{align*}
1x−1y=  \begin{align*}\left (\frac{1}{x} \right ) \left (\frac{1}{y}\right ) = \end{align*}
 \begin{align*}\frac{\frac{1}{x}}{\frac{1}{y}} = \end{align*}

\begin{align*}\frac{x^2 + 2x + 1}{x^2  1}\end{align*}
 Simplify
 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.
 Are there any \begin{align*}x\end{align*}values for which the new expression does not equal the old expression?
 \begin{align*}\frac{2}{x^2  9}  \frac{4}{x^2 + 2x  15} = \end{align*}

\begin{align*}\frac{4x^2  25 }{x^2 + 2x + 1} \times \frac{x^2 + 4x + 3}{2x^2 + x  15}\end{align*}
 Multiply
 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!)
 Are there any \begin{align*}x\end{align*}values for which the new expression does not equal the old expression?
Name: __________________
Homework—Rational Expressions

\begin{align*}\frac{x^2  6x + 5}{x^2  16} \times \frac{x^2 + 8x + 16}{x^2  7x + 10}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{\frac{2x^2 + 7x + 3}{x^3 + 4x}} {\frac{x^2 + x  6}{3x}}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{1}{x  1}  \frac{1}{x + 1}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
 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?

\begin{align*}\frac{x  3}{x^2 + 9x + 20}  \frac{x  4}{x^2 + 8x + 15}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{x + 1}{4x^2  9} + \frac{4x}{6x^2  9x}\end{align*}
 Simplify
 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

\begin{align*}\frac{1}{x  1}  \frac{1}{2x}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
 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?

\begin{align*}\frac{3x^2 + 5x  8}{6x + 16} \times \frac{4x^3 + x}{3x^3  3x}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{2}{4x^3  x} + \frac{3}{2x^3 + 3x^2 + x}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{\frac{x^2 + 9}{x^2  4}} {\frac{x^2 + 7x + 12}{x^2 + 2x  8}}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?
 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?

\begin{align*}\frac{x}{x^2  25} + <\mathrm{something}> = \frac{x^2 + 1}{x^3  9x^2 + 20x}\end{align*}
 What is the something?
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{x  6}{x  3} = \frac{x + 18}{2x + 7}\end{align*}
 Solve for \begin{align*}x\end{align*}. You should get two answers.
 Check by plugging one of your answers back into the original equation.
Name: __________________
Dividing Polynomials
 \begin{align*}\frac{28k^3 p  42 kp^2 + 56kp^3}{14 kp} = \end{align*}

\begin{align*}\frac{x^2  12x  45}{x + 3} =\end{align*}
 \begin{align*}\frac{2y^2 + y  16}{y  3} = \end{align*}
 Test your answer by choosing a number for \begin{align*}y\end{align*} and seeing if you get the same answer.
 \begin{align*}\frac{2h^3  5h^2 + 22h + 51}{2h + 3} = \end{align*}
 \begin{align*}\frac{2x^3  4x^2 + 6x  15}{x^2 + 3} = \end{align*}

\begin{align*}\frac{x^3  4x^2}{x  4} =\end{align*}
 \begin{align*}\frac{x^3  27}{x  3} = \end{align*}
 Test your answer by multiplying back.
 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

\begin{align*}\frac{x  3}{x^2 + 9x + 20}  \frac{x  4}{x^2 + 8x + 15}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{2}{x^2  1} + \frac{x}{x^2  2x + 1}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{4x^3  9x}{x^2  3x  10} \times \frac{2x^2  20x + 50}{6x^2  9x}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{\frac{1}{x}}{\frac{x  1}{x^2}}\end{align*}
 Simplify
 What values of \begin{align*}x\end{align*} are not allowed in the original expression?
 What values of \begin{align*}x\end{align*} are not allowed in your simplified expression?

\begin{align*}\frac{x  1}{2x  1} = \frac{x+7}{7x + 4}\end{align*}
 Solve for \begin{align*}x\end{align*}.
 Test one of your answers and show that it works in the original expression. (No credit unless you show your work!)

\begin{align*}\frac{6x^3  5x^2  5x + 34}{2x + 3}\end{align*}
 Solve by long division.
 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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