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8.1: Rational Expressions

Difficulty Level: At Grade Created by: CK-12

Name: __________________

Rational Expressions

1. $\frac{1}{x} + \frac{1}{y}$
2. Check your answer by plugging $x=2$ and $y=4$ into both my original expression, and your simplified expression. Do not use calculators or decimals.
2. $\frac{1}{x} - \frac{1}{y} =$
3. $\left (\frac{1}{x} \right ) \left (\frac{1}{y}\right ) =$
4. $\frac{\frac{1}{x}}{\frac{1}{y}} =$
5. $\frac{x^2 + 2x + 1}{x^2 - 1}$
1. Simplify
2. Check your answer by plugging $x=3$ into both my original expression, and your simplified expression. Do not use calculators or decimals.
3. Are there any $x-$values for which the new expression does not equal the old expression?
6. $\frac{2}{x^2 - 9} - \frac{4}{x^2 + 2x - 15} =$
7. $\frac{4x^2 - 25 }{x^2 + 2x + 1} \times \frac{x^2 + 4x + 3}{2x^2 + x - 15}$
1. Multiply
2. 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!)
3. Are there any $x-$values for which the new expression does not equal the old expression?

Name: __________________

Homework—Rational Expressions

1. $\frac{x^2 - 6x + 5}{x^2 - 16} \times \frac{x^2 + 8x + 16}{x^2 - 7x + 10}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
2. $\frac{\frac{2x^2 + 7x + 3}{x^3 + 4x}} {\frac{x^2 + x - 6}{3x}}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
3. $\frac{1}{x - 1} - \frac{1}{x + 1}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
4. 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?
4. $\frac{x - 3}{x^2 + 9x + 20} - \frac{x - 4}{x^2 + 8x + 15}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
5. $\frac{x + 1}{4x^2 - 9} + \frac{4x}{6x^2 - 9x}$
1. Simplify
2. 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 $\frac{x}{36} = \frac{15}{36}$. What are all the values $x$ 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 $\frac{x}{18} = \frac{15}{36}$. What are all the values $x$ can take that make this statement true?

Hey, $x$ came out being a fraction. Can he do that?

Umm, yeah.

OK, one more pretty easy one.

3. $\frac{x^2 + 2}{21} = \frac{9}{7}$

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. $\frac{x + 2}{x + 3} = \frac{x + 5}{x + 4}$

5. $\frac{2x + 6}{x + 3} = \frac{x + 5}{2x + 7}$

6. $\frac{x + 3}{2x - 3} = \frac{x - 5}{x - 4}$

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. $\frac{1}{x - 1} - \frac{1}{2x}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
4. 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?
2. $\frac{3x^2 + 5x - 8}{6x + 16} \times \frac{4x^3 + x}{3x^3 - 3x}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
3. $\frac{2}{4x^3 - x} + \frac{3}{2x^3 + 3x^2 + x}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
4. $\frac{\frac{x^2 + 9}{x^2 - 4}} {\frac{x^2 + 7x + 12}{x^2 + 2x - 8}}$
1. Simplify
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
4. 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?
5. $\frac{x}{x^2 - 25} + <\mathrm{something}> = \frac{x^2 + 1}{x^3 - 9x^2 + 20x}$
1. What is the something?
2. What values of $x$ are not allowed in the original expression?
3. What values of $x$ are not allowed in your simplified expression?
6. $\frac{x - 6}{x - 3} = \frac{x + 18}{2x + 7}$
1. Solve for $x$. You should get two answers.
2. Check by plugging one of your answers back into the original equation.

Name: __________________

Dividing Polynomials

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

Name: __________________

Sample Test: Rational Expressions

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

Extra credit: I am thinking of two numbers, $x$ and $y$, 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 $x$, can you find a general formula that will always give me the other one?

c. Is there any number $x$ that has no possible $y$ to work with?

Feb 23, 2012

Apr 29, 2014