lesson 4 add, subtract, multiply, and divide rational expressions
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What is a rational expression?
• A rational expression is an expression that can be written as a ratio of two polynomials.
Ex.
8
-2x
x
x2 – 3x
x + 5
x² + 2x - 15
Excluded values
• A rational expression is undefined when the denominator is 0. A number that makes the denominator 0 is an excluded value.
• Let’s look at the examples before…are there any values that should be excluded?
8
-2x
x
x2 –3x
x + 2
x² + 2x - 15
Examples
• Find the excluded values, if any, of the expression.
1. 9x
5x - 15
2. x -1
x² - 16
3. x + 6
x² + 4x - 12
To find the excluded values, you have to factor the denominator (if possible). Then set each factor equal to ZERO and solve.
Simplest Form
• A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1.
• Examples: Are these expressions in simplest form?
8
-2x
x
3x - 9
x + 5
x² + 2x - 15
What about these?
• Simplify, if possible. State the excluded values.
2m
8m(m-1)
11
y + 6
7q² - 14 q
14q²
Try these…
• Simplify and state the excluded values.
x² + 4x – 21
x² - 5x + 6
x² + 13x + 42
x² -2x -63
Multiplying Rational Expressions
• ½ * ¾ =
• 2/5 * 5/4 =
• 2/3 * 5/4 =
• 2x² * 1/x =
• 4xy/5 * 20y/4x²
Multiply and Divide Rational Expressions with variables
• Factor the numerators and denominators.• Simplify where possible (…Cancel)• Multiply the numerators and denominators straight
across.• List restrictions (excluded values) if any…
Example: x² + x – 6 5x² + 15x
10x² - 20x x² - 2x - 15*
So…to divide rational expressions…
1. Find the reciprocal of the second fraction.
2. Simplify according to your rules…3. Multiply straight across…
Graphic Organizer
• Fill in your graphic organizer “How do you multiply or divide rational expressions?”
• Work each example in the space provided.
Practice
• Complete textbook pp. 441-444
• Dividing rational expressions handout.
• Multiplying rational exp handout.
Finding the LCM
• Find the LCM of 6x and 8x²
Step 1: Write the factors of each expression.
6x = 2 * 3 * x
8x² = 2 * 2 * 2 * x * x
Next…
• Step 2:
6x = 2 * 3 * x
8x² = 2 * 2 *2 * x * x
• Step 3: List the common factors once…and then list every other factor…
• Step 4: MULTIPLY
2 * x * 3 * 2 * 2 * x =24 x²
Circle the factors that both
expressions have in common…
Find the LCM of…
• x² + 2x – 8 and x² + 7x + 12Step 1: List the factors of each polynomialx² + 2x – 8 = (x-2)(x+4)x² + 7x + 12 = (x+3)(x+4)Step 2: Circle the common factors.Step 3:List the common factor once and then
the other factors…then multiply…(x+4)(x-2)(x+3) There’s no need to multiply this out…Leave it as it is…
Add/Subtract Rationals
• x + 3 + x – 2
7x 7x
• 5x + 7 - 2x – 9
3x – 4 3x – 4
So, on these I need to
add/subtract the numerators and just bring
over the denominator,
then simplify, if possible?
What if the denominators are not the same?
• 11 + 15 12x² 16x5
1. Find the LCD of the denominators…aka LCM.
2. Multiply top and bottom by missing factors…
3. Now that the denominators are the same…follow your steps for adding fractions…
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