8.1: Rational Expressions
Name: __________________
Rational Expressions

1x+1y  Add
 Check your answer by plugging
x=2 andy=4 into both my original expression, and your simplified expression. Do not use calculators or decimals.

1x−1y= 
(1x)(1y)= 
1x1y= 
x2+2x+1x2−1  Simplify
 Check your answer by plugging
x=3 into both my original expression, and your simplified expression. Do not use calculators or decimals.  Are there any
x− values for which the new expression does not equal the old expression?

2x2−9−4x2+2x−15= 
4x2−25x2+2x+1×x2+4x+32x2+x−15  Multiply
 Check your answer by plugging
x=−2 into both my original expression, and your simplified expression. (If they don’t come out the same, you did something wrong!)  Are there any
x− values for which the new expression does not equal the old expression?
Name: __________________
Homework—Rational Expressions

x2−6x+5x2−16×x2+8x+16x2−7x+10  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?

2x2+7x+3x3+4xx2+x−63x  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?

1x−1−1x+1  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?  Test your answer by choosing an
x value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?

x−3x2+9x+20−x−4x2+8x+15  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?

x+14x2−9+4x6x2−9x  Simplify
 Test your answer by choosing an
x 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
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
Hey,
Umm, yeah.
OK, one more pretty easy one.
3.
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.
5.
6.
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

1x−1−12x  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?  Test your answer by choosing an
x value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?

3x2+5x−86x+16×4x3+x3x3−3x  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?

24x3−x+32x3+3x2+x  Simplify
 What values of
x are not allowed in the original expression?  What values of
x are not allowed in your simplified expression?

x2+9x2−4x2+7x+12x2+2x−8  Simplify
 What values of
x are not allowed in the original expression?  What values of
x 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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