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Rational Expressions. Multiplying / Dividing. Let’s first review how we multiply and divide fractions. Multiplying / Dividing. When multiplying/ dividing, do we have to have a common denominator? Nope. Multiplying / Dividing. Is there anything special that we have to do? Nope. - PowerPoint PPT Presentation
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Rational Expressions
Multiplying / Dividing
Let’s first review how we multiply and divide fractions.
Multiplying / Dividing
When multiplying/ dividing, do we have to have a common denominator?
Nope
Multiplying / Dividing
Is there anything special that we have to do?
Nope
Multiplying / Dividing
How do we multiply fractions?
Multiply numerators and denominators.
Multiplying / Dividing
Examples:4x3y
5x 3
?
9a(b 3c)
a 34
?
Multiplying / Dividing
Examples:4x3y
5x 3
20x2
9y9a
(b 3c)a 34
9a2 27a4b 12c
Multiplying / Dividing
Let’s talk about simplifying.
When can we cancel things in the numerator and denominator?
Multiplying / Dividing
You must have exactly the same factor.
x - 3 can only be simplified by x - 3, not by x, not by 3, only by x - 3.
Multiplying / Dividing
Simplify: 3(2x 7)(x 5)9(x 5)(7x 2)
?
Multiplying / Dividing
Simplify: 3(2x 7)(x 5)9(x 5)(7x 2)
2x 73(7x 2)
3
Multiplying / Dividing
Simplify: 2x 10x 5
Multiplying / Dividing
Simplify: factor first2x 10x 5
2(x 5)x 5
Multiplying / Dividing
Simplify: cancel2x 10x 5
2(x 5)x 5
2
Multiplying / Dividing
Simplify: factor, then cancel
6b3 24b2
b2 b 20
Multiplying / Dividing
Simplify: factor, then cancel6b3 24b2
b2 b 20
6b2 (b 4)(b 5)(b 4)
6b2
b 5
Multiplying / Dividing
Complete wkst 107Completely simplify each polynomial (numerator and denominator), then cancel.
Multiplying / Dividing
When multiplying rational expressions, factor each numerator and denominator first, then cancel, then multiply (squish them together).
Multiplying / Dividing
Steps to multiply:1)factor completely2)cancel3)multiply (squish)
Multiplying / Dividing
Multiply:
We have nothing to factor so just cancel and multiply.
x3
2y26y4
xy
Multiplying / Dividing
Multiply:x3
2y26y4
xy3x2y1
3x2y3x2 y2
y
Multiplying / Dividing
Multiply:2x2 5x 7x 4
x 2 4xx2 2x 1
Multiplying / Dividing
Factor:(2x 7)(x 1)
x 4
x(x 4)(x 1)(x 1)
Multiplying / Dividing
Cancel:(2x 7)(x 1)
x 4
x(x 4)(x 1)(x 1)
Multiplying / Dividing
Multiply:(2x 7)1
x
(x 1)x(2x 7)x 1
Multiplying / Dividing
Realize: sometimes you may see people go ahead and multiply (such as the last numerator), you don’t have to do this for me. Just be able to recognize it if you see it.
Multiplying / Dividing
Work on wkst 111
Multiplying / Dividing
Now on to dividing.This is exactly like multiplying, except for ONE step.
Multiplying / Dividing
How do we divide fractions?
We multiply by the reciprocal (inverse) of the divisor (2nd fraction).
34
14
34
41
124
3
Multiplying / Dividing
Divide:x2 x 12x2 11x 24
x2 2x 8x2 8x
Multiplying / Dividing
Divide: first step is flip & multiply.x2 x 12x2 11x 24
x2 8xx2 2x 8
Multiplying / Dividing
Divide: Now proceed like a multiplication problem. Factor first, cancel, multiply.(x 4)(x 3)(x 3)(x 8)
x(x 8)
(x 4)(x 2)
Multiplying / Dividing(x 4)(x 3)
(x 3)(x 8)
x(x 8)(x 4)(x 2)
(x 4)(x 3)(x 3)(x 8)
x(x 8)
(x 4)(x 2)
Multiplying / Dividing
Simplifyx
(x 2)
Multiplying / Dividing
Wkst 113/112Start on the dividing side first.
Adding/Subtracting
What do we have to have in order to add or subtract fractions?
Right a common denominator.
Adding/Subtracting
when we talk about CDs, we mean denominators that contain the same factors.
To find our CD, we will first factor the ones we have.
Adding/Subtracting
Then we will multiply each denominator by the factors it is missing to create a CD.
Remember, we must also multiply the numerator by that same factor.
Adding/Subtracting
Find the common denominator for these two rational expressions.2x5ab3
4y3a2b2
Adding/Subtracting2x
5ab3 4y3a2b2
2x(3a)5ab3(3a)
4y(5b)3a2b2 (5b)
Adding/Subtracting2x(3a)
5ab3(3a)4y(5b)3a2b2 (5b)
6ax15a2b3
20by15a2b3
Adding/Subtracting
Now that we have a CD, we just simplify (we did already) and add the numerators.
6ax 20by15a2b3
Adding/Subtracting
If we could factor the numerator and denominator, we would to see if we could simplify more.
Steps to Add/Subtract
Factor denominatorsFind CD & equivalent fractions
Simplify/add/sub numer.Factor numer/denomCancel
Adding/Subtracting
Simplify:x
x2 5x 6
2x 2 4x 4
Adding/Subtracting
Factorx
(x 2)(x 3)
2(x 2)(x 2)
Adding/Subtracting
Find CDWe need a factor of (x+2) in the first denominator and a factor of (x+3) in the second denominator.
Adding/Subtracting
Create equivalent fractions.x(x 2)
(x 2)2(x 3)
2(x 3)(x 2)2 (x 3)
Adding/Subtracting
Simplify the numerators.x 2 2x
(x 2)2(x 3)
2x 6(x 2)2 (x 3)
Adding/Subtracting
Subtract.(x2 + 2x) - (2x + 6) = x2 - 6
x2 6(x 2)2(x 3)
Adding/Subtracting
Can we factor the numerator? No, we are done. x2 6
(x 2)2(x 3)
Adding/Subtracting
Simplify:
x 52x 6
x 74x 12
Adding/Subtractingx 5
2(x 3)x 74(x 3)
2(x 5)4(x 3)
x 74(x 3)
Adding/Subtracting2x 10
4(x 3)x 74(x 3)
x 34(x 3)
14
Adding/Subtracting
pg. 57321- 37 odd
Adding/Subtracting
pg. 57321- 37 odd
Complex FractionsComplex fractions are those fractions whose numerator & denominator both contain fractions.
To simplify them, we just multiply by the CD.
Complex FractionsExample:What would
be the CD?6Multiply every
term by 6.
x x3
x x6
Complex FractionsSimplify.
6x 6x3
6x 6 x6
6x 2x6x x
Complex FractionsSimplify.
6x 2x6x x
8x5x
85
Complex FractionsSimplify.CD is ?xy
2xy
1
2xy
yx
Complex FractionsMultiply each term by xy.
xy2xy
1(xy)
(xy)2xy
(xy)
yx
Complex FractionsSimplfy.
xy2xy
1(xy)
(xy)2xy
(xy)
yx
Complex FractionsFinal answer.
2x2 xy2x2 y2
Complex FractionsSteps to simplify.Find the CDMultiply EACH term by the CD
Cancel and Simplify
Complex FractionsWork on Complex Fraction Worksheet
Rational EquationsSolving Rational Equations would be easy, except for the rational part.
How can we get rid of the the fractions?
Rational EquationsMultiply BOTH SIDES by the common denominator
This will be very similar to adding/subtracting rational expressions.
Rational EquationsOnce you have multiplied by the CD, just solve the equation like you would normally. Use your calculator if you want to.
Rational EquationsSolve:
24r 3
36r 3
Rational EquationsWhat is the CD?(r - 3)(r + 3)
24r 3
36r 3
Rational EquationsMultiply by (r - 3)(r + 3) on both sides.24(r 3)(r 3)
r 336(r 3)(r 3)
r 3
Rational EquationsSimplify & distribute24(r 3)(r 3)
r 336(r 3)(r 3)
r 324(r 3) 36(r 3)24r 72 36r 108
Rational EquationsSolve24r 72 36r 108180 12r15 r
Rational EquationsSolve:x 13(x 2)
5x6
1x 2
Rational EquationsCommon Denominator is6(x-2)x 13(x 2)
5x6
1x 2
Rational EquationsMultiply by 6(x-2)6(x 2)(x 1)3(x 2)
(5x)(6)(x 2)6
1(6)(x 2)x 2
Rational EquationsCancel6(x 2)(x 1)3(x 2)
(5x)(6)(x 2)6
1(6)(x 2)x 2
2
Rational EquationsSimplify2(x 1) 5x(x 2) 62x 2 5x2 10x 6
0 5x 2 2x 10x 6 25x2 12x 4 0
Rational EquationsSolve5x2 12x 4 0
5x2 10x 2x 4 0(5x2 10x) ( 2x 4) 0
Rational EquationsSolve5x(x 2) 2(x 2)0(5x 2)(x 2) 0
x 25,2
Rational EquationsTry some on your ownpg. 5816 - 9 => solve them all
Rational EquationsAssignmentpg. 58213 - 18 all (just solve), 25 - 28 all