mutiplyin and dividing expressions
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Simplifying Multiplying Dividing
Grade 9 B – T3 -W 5- 13/14
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Simplifying
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Remember, denominators can not = 0.
Now,lets go through the steps to simplify a rational expression.
Examples of rational expressions
2
4 8 4 7, ,3 3 5 9
x yx x y y
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Simplify: 7x 7x2 1
Step 1: Factor the numerator and the denominator completely looking for common factors.
7x 7 7(x 1)
x2 1 (x 1)(x 1)Next
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7x 7x2 1
7(x 1)
(x 1)(x 1)
What is the common factor?x 1
Step 2: Divide the numerator and denominator by the common factor.
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7(x 1)
(x 1)(x 1)
7(x 1)(x 1)(x 1)
1
1
Step 3: Multiply to get your answer.
Answer: 7
x 1
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Looking at the answer from the previous example, what value of x would make the denominator 0?
x= -1
The expression is undefined when the values make the denominator equal to 0
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How do I find the values that make an expression undefined?
Completely factor the original denominator.
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Ex: 2ab(a 2)(b 3)
3ab(a2 4)
3ab(a2 4) 3ab(a 2)(a 2)
The expression is undefined when: a= 0, 2, and -2 and b= 0.
Factor the denominator
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Lets go through another example.
3a3 a4
2a3 6a2
3a3 a4
2a3 6a2 a3 (3 a)
2a2 (a 3)
Factor out the GCF
Next
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3
22 ( 3)(3 )a
a aa
3 factored is 1( 3)a a
cancel like factors3
2
1 ( 3)2 ( 3)
a aa a
1
1
3
2
1( 3)2 ( 3)
aa a
a
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3
2
12
aa
2cancel out the like factor a
12a
1
a
answer
What values is the original expression undefined?
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Now try to do some on your own.
2
2
3 2
3 2
5 61) 9
5 102) 6 16
x xxx x
x x x
Also find the values that make each expression undefined?
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Multiplying and
Dividing
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Multiplying and Dividing Rational Expressions
Remember that a rational number can be expressed as a quotient of two integers. A rational expression can be expressed as a quotient of two polynomials.
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Remember how to multiply fractions:
First you multiply the numerators then multiply the denominators.
5 2:6 20
Ex 10 1120 12
5 26 20
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The same method can be used to multiply rational expressions.
Ex: 4a2
5ab3 3bc
12a3 4 a a 3 bc5 a b b b 12 a a a
11 1 1 1
1 1 1 1
c
5b2 a2
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Let’s do another one.
Ex: x3 3x2
x2 5x 6
x2 10x 9x2 6x 27
Step #1: Factor the numerator and the denominator.
x2 (x 3)(x 6)(x 1)
(x 1)(x 9)(x 9)(x 3)
Next
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Step #2: Divide the numerator and denominator by the common factors.
x2 (x 3)(x 6)(x 1)
(x 1)(x 9)(x 9)(x 3)1
1
1
1
1
1
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Step #3: Multiply the numerator and the denominator.
x2
x 6
Remember how to divide fractions?
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Multiply by the reciprocal of the divisor. 4
5
1625
45
2516
4 25516
1
1
5
4
54
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Dividing rational expressions uses the same procedure.
Ex: Simplify
y 2y2 10 y 24
y2 2y
y2 2y 8
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y 2y2 10 y 24
y2 2y
y2 2y 8
y 2y2 10 y 24
y2 2y 8
y2 2y
y 2(y 12)(y 2)
(y 4)(y 2)
y(y 2)
1 1
1 1
Next
4( 12)y
y y
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Now you try to simplify the expression:
x 3x2 4x 12
2x2 6x
x 2
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Choose the correct answer .
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Answer: 1
2x(x 6)
Now try these on your own.
1) x + 3
2x3 2x2 x2 7x 6
x2 10x 21
2) 3x 67x 7
5x 1014x 14