Rational Expressions l Multiplying/Dividing l Adding/Subtracting l Complex Fractions.

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!