Simplifying
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Transcript of Simplifying
Simplifying
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.
Remember, denominators cannot = 0.
Now, let’s go through the steps to simplify a rational expression.
Examples of rational expressions
2
4 8 4 7, ,
3 3 5 9
x y
x x y y
Simplify: 7x 7
x2 1
Step 1: Factor the numerator and the denominator completely looking for common factors:
Numerator:
Denominator:
7x 7 7(x 1)
x2 1 (x 1)(x 1)
7x 7
x2 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.
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
Looking at the answer from the previous example …
what value of x would make the denominator 0?x = (-1) makes the denom=0
The expression is undefined when the values make the denominator equal to 0
How do I find the values that make an expression undefined?
Completely factor the original denominator.
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
Lets go through another example.
3a3 a4
2a3 6a2
3a3 a4
2a3 6a2 a3 (3 a)
2a2 (a 3)
Factor out the GCF
Next
3
22 ( 3)
(3 )a
a a
a
3 factored is 1( 3)a a
cancel like factors3
2
1 ( 3)
2 ( 3)
a a
a a
1
1
3
2
1( 3)
2 ( 3)
a
a a
a
3
2
1
2
a
a
2cancel out the like factor a
1
2
a
1
a
answer
What values is the original expression undefined?
Now try to do some on your own.
2
2
3 2
3 2
5 61)
9
5 102)
6 16
x x
x
x x
x x x
Also find the values that make each expression undefined??
2
2
3 2
3 2
5 61)
9
5 102)
6 16
x x
x
x x
x x x
1) x² - 5x + 6 = (x – 2)(x – 3) x² - 9 (x + 3)(x – 3)
Cancel out both (x – 3)’s
= (x – 2) (x + 3)
2) 5x³ + 10x² = 5x²(x + 2)
x³ - 6x² - 16x x(x² - 6x – 16)
= 5x²(x + 2) x(x – 8)(x + 2)
Cancel out both (x + 2)’s
= 5x² = 5x x(x – 8) (x – 8)
Do you remember how to multiply fractions??
First you multiply the numerators then multiply the denominators.
5 2:6 20
Ex 10 1
120 12
5 2
6 20
The same method can be used to multiply rational expressions.
Ex: 4a2
5ab3 3bc
12a3 4 a a 3 bc
5 a b b b 12 a a a
11 1 1 1
1 1 1 1
c
5b2 a2
Let’s do another one.
Ex: x3 3x2
x2 5x 6
x2 10x 9
x2 6x 27Step #1: Factor the numerator and the denominator.
x2 (x 3)
(x 6)(x 1)(x 1)(x 9)
(x 9)(x 3)Next
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
Step #3: Multiply the numerator and the denominator.
x2
x 6
Remember how to divide fractions?
Multiply by the reciprocal of the divisor.
4
5
16
25
4
525
16
4 25
516
1
1
5
4
5
4
Dividing rational expressions uses the same procedure.
Ex: Simplify
y 2
y2 10 y 24
y2 2y
y2 2y 8
y 2
y2 10 y 24
y2 2y
y2 2y 8
y 2
y2 10 y 24y2 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
Now you try to simplify the expression:
x 3
x2 4x 12
2x2 6x
x 2
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
Here are the answers:
1) x 6
2x2 (x 7)
2) 6(x 1)5(x 1)
Now wasn’t that fun???!!!
Your homework is: