9.4 multiply and divide rational expressions vocabulary simplified form of a rational expression
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9.4 Multiply and Divide Rational ExpressionsVOCABULARY
Simplified form of a rational expression
VOCABULARYSimplified form of a rational expression
A rational expression in which its numerator and denominator have no common factors (other than ±1 )
SIMPLIFYING RATIONAL EXPRESSIONS
Step 1: Factor numerator and denominator
Step 2: Divide out common factors
Step 3: Simplified form
Example 1Simplify a rational expression(x² + 7x + 10) (x² - 4)Step 1: Factor numerator and denominator(x² + 7x + 10) = (x+2)(x+5) (x² - 4) (x+2)(x-2)Step 2: Divide out common factor(x+2)(x+5)(x+2)(x-2)
Step 3: Simplified Form
(x+5)
(x-2)
CheckpointSimplify the expression
1. x² - 2x – 15
x² + 4x + 3
(x – 5)(x + 3)
(x + 1)(x + 3)
(x – 5)
(x + 1)
Practice!
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MULTIPLYING RATIONAL EXPRESSIONS
Step 1: Factor numerators and denominators
Step 2: Multiply numerators and denominators
Step 3: Divide out common factors
Step 4: Simplified form
Example 2Multiply rational expressions2x² + 4x x² - 9x + 18x² - 4x – 12 * 2xStep 1: Factor 2x(x + 2) (x – 3)(x – 6)(x + 2)(x – 6) * 2xStep 2: Multiple 2x(x + 2)(x – 3)(x – 6) (x + 2)(x – 6)2x
2x² + 4x x² - 9x + 18x² - 4x – 12 * 2x
Step 3: Divide
2x(x + 2)(x – 3)(x – 6)
(x + 2)(x – 6)2x
Step 4: Simplify
x – 3
CheckpointMultiply the expression6x² + 18x x² - x – 2x² + x – 6 * x² - 7x – 86x(x + 3) (x-2)(x+1)(x+3)(x-2) * (x-8)(x+1)6x(x + 3)(x-2)(x+1)(x+3)(x-2)(x-8)(x+1) 6x (x-8)
Example 3Multiply a rational expression by a polynomial
x – 4 * (x² - x + 1)x³ + 1x – 4 * (x² - x + 1)x³ + 1 1(x-4)(x² - x + 1)(x+1)( x² - x + 1)X – 4 X + 1
Dividing Rational ExpressionsStep 1: Multiply by reciprocal
Step 2: Factor
Step 3: Divide
Step 4: Simplify
Example 4Divide rational expressions 3 ● 8x² - 8x x + 7 ● x² + 6x – 7Step 1: Multiply by reciprocal 3 x² + 6x – 7x+7 * 8x² - 8x Step 2: Factor 3 (x+7)(x-1)(x+7)(8x)(x-1)
3 ● 8x² - 8x x + 7 ● x² + 6x – 7 cont.
Step 3: Divide
3 (x+7)(x-1)
(x+7)(8x)(x-1)
Step 4: Simplify
3__
8x
CheckpointDivide the expression3. x – 5 ● 2x² - 11x + 5 9x² - 18x ● 2x² - 5x + 2 (x – 5) (2x² - 5x + 2) (9x² - 18x)(2x² - 11x + 5)(x – 5)(2x-1)(x-2)9x(x–2)(2x-1)(x-5) 1_9x
A farmer wants to fence in a square field
that measures 2x on each side.
Write a simplified rational
expression for the ratio of the
field’s perimeter to its area.
From 1987 to 1996 the total acres of farmland L ( in millions) and the total number of farms F ( in hundreds of thousands ) in the United States can be modeled by:
t represents the number of years since 1987. Write a model for the average number of acres per
farm as a function of the year. What was the average number of acres per farm in
1993?
105.
20.2101.
10482.
9993.43
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Warm-up
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