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Rational Expressions
Multiplying/DividingAdding/SubtractingComplex Fractions
Multiplying / Dividing
Let’s first review how we multiply and divide ordinary fractions.
Do we need a common denominator?
No!
Multiplying / Dividing
How do we multiply ordinary fractions?•Multiply across: numerators times numerators and denominators times denominators.
Multiplying / Dividing
How do we multiply rational expressions with variables?•Multiply across: numerators times numerators and denominators times denominators.
Multiplying / Dividing
For Example:
4x3y
•5x−3
=20x2
−9y
9a(b−3c)
•a+3
4=
9a2 +27a4b−12c
Multiply AcrossUsing Power Rules!
Multiplying / Dividing
When do we cancel things in fractions? We can only cancel identical factors that
appear in both the numerator and denominator. • (x - 3) can only be cancelled by
(x - 3), not by x, not by 3.
Multiplying / Dividing
3(2x−7)(x +5)9(x+5)(7x−2)
=
2x−73(7x−2)
3
Here, 3, 9, (x+5) and (x + 5) are all identical factors and can be cancelled.
(2x-7) and (7x-2) are factors, but they aren’t identical; we can’t cancel any part of them!
Multiplying / Dividing
To simplify: 2x+10x+5
We must factor first, then we can cancel:
2x+10x+5
=2(x+5)
x+5=2
Simplify: factor, then cancel
6b3 −24b2
b2 +b−20=
6b2(b−4)(b+5)(b−4)
=6b2
b+5
Multiplying / Dividing
Multiplying / Dividing
When multiplying rational expressions• factor each numerator and
denominator first• then cancel identical factors• then multiply across: numerators by
numerators and denominators by denominators.
Multiplying / Dividing
Practice:
2x2 +5x−7x +4
•x2 +4x
x2 −2x+1
Multiplying / Dividing
2x2 +5x−7x +4
•x2 +4x
x2 −2x+1
(2x+7)(x −1)x+4
•x(x+4)
(x−1)(x−1)=
(2x+7)(x −1)x+4
•x(x+4)
(x−1)(x−1)=
(2x+7)1
•x
(x−1)=
x(2x +7)x−1
=
Solution:
Multiplying / Dividing
Now on to dividing. This is exactly like multiplying,
except for ONE step. We multiply by the reciprocal of
the 2nd fraction!
Multiplying / Dividing
Divide:x2 −x−12
x2 +11x +24÷
x2 −2x −8x2 +8x
Change it to multiplication and flip the 2nd fraction:
x2 −x−12x2 +11x +24
•x2 +8x
x2 −2x −8
Multiplying / Dividing
Divide: Now proceed like a multiplication problem. Factor first, cancel, multiply.
x2 −x−12x2 +11x +24
•x2 +8x
x2 −2x −8
=(x−4)(x+3)(x +3)(x+8)
•x(x+8)
(x−4)(x+2)
x(x+2)
=
Adding/Subtracting
What do we have to do to add or subtract ordinary fractions?•Change one or both fractions so
they have the same common denominator.
Find the LCD for two fractions with monomial denominators:
2x
5ab3 +4y
3a2b2
Adding/Subtracting
The key is that the LCD be something we can reach by multiplying each denominator by missing terms.
If we multiply the 1st denominator by 3a we get:
Adding/Subtracting
2x 3a( )5ab3 3a( )
=6xa
15a2b3
4y 5b( )3a2b2 5b( )
=20yb
15a2b3
If we multiply the 2nd denominator by 5b we get:
Same den. (LCD)
Once we have the same denominator, we add the numerators:
Adding/Subtracting
6xa15a2b3 +
20yb15a2b3 =
6xa+20yb15a2b3
After adding the numerators, try to factor and cancel in the final fraction if possible.
Find the LCD for two fractions with polynomial denominators:
Adding/Subtracting
First we must factor the denominators...
x
x2 + 5x+ 6−
2x2 + 4x+ 4
The LCD will need to include at least :• One (x+2) factor from the 1st fraction• One (x+3) factor from the 1st fraction• Two (x+2) factors from the 2nd fraction
Adding/Subtracting
x
(x + 2)(x+ 3)−
2(x+ 2)(x+ 2)
We don’t need three (x+2) terms, two will satisfy the needs of BOTH fractions!
Adding/Subtracting
Get the LCD: (x+2)(x+2)(x+3)
x(x + 2)(x+ 2)2(x+ 3)
−2(x+ 3)
(x+ 2)2 (x+ 3)
x
(x + 2)(x+ 3)−
2(x+ 2)(x+ 2)
(x+2)
(x+2)
(x+3)
(x+3)
x 2 + 2x(x+ 2)2(x+ 3)
−2x+ 6
(x+ 2)2 (x+ 3)
Adding/Subtracting
Subtract the numerators:
x 2 + 2x(x+ 2)2(x+ 3)
−2x+ 6
(x+ 2)2 (x+ 3)x2 −6
(x+ 2)2(x+ 3)=
We cannot factor the numerator, so we are finished (don’t try to cancel anything).
Adding/Subtracting
Practice:
x−52x−6
−x−7
4x−12
Adding/Subtracting
Practice:
Factor: x−52(x−3)
−x−7
4(x−3)
LCD must contain at least: a multiple of 2, a multiple of 4, a factor of (x-3).
x−52(x−3)
−x−7
4(x−3)2*
2*
Adding/Subtracting
2x −104(x−3)
−x−7
4(x−3)=
x−34(x−3)
=14
Subtract:
Here, we do have factors to cancel:
Complex Fractions
Complex fractions are those fractions whose numerators &/or denominators contain fractions.•To simplify them, we just multiply
the top & bottom by the LCD.
Complex Fractions
Example
x+x3
x−x6
What would the LCD be? The denominators are 3 and 6, the LCD is 6.
Complex Fractions
Multiply the top & bottom both by 6:
x+x3
⎛ ⎝
⎞ ⎠
x−x6
⎛ ⎝
⎞ ⎠
*6
*6
=6x+6⋅
x3
6x−6⋅x6
=6x +2x6x−x
=8x5x
=85
2
Complex Fractions
Simplify: 2xy
+1
2xy
+yx
Complex Fractions
What would the LCD be?
2xy
+1
2xy
+yx
The denominators are y and x, the LCD is xy.
Complex Fractions
Multiply top & bottom by LCD:
xy⋅2xy
+1⎛ ⎝ ⎜
⎞ ⎠ ⎟
xy⋅2xy
+yx
⎛ ⎝ ⎜
⎞ ⎠ ⎟
=xy⋅
2xy
+xy⋅1
xy⋅2xy
+xy⋅yx
=2x2 +xy2x2 +y2
Complex Fractions
The final answer is:
2x2 +xy2x2 +y2
We cannot cancel any terms in this fraction!
