Adding and Subtracting Rational Expressions Goal 1 Determine the LCM of polynomials Goal 2 Add and Subtract Rational Expressions
Feb 24, 2016
Adding and Subtracting Rational Expressions
Goal 1 Determine the LCM of polynomials
Goal 2 Add and Subtract Rational Expressions
What is the Least Common Multiple?
Least Common Multiple (LCM) - smallest number or
polynomial into which each of the numbers or polynomials will divide evenly.
Fractions require you to find the Least Common Multiple (LCM) in order to add and subtract them!
The Least Common Denominator is the LCM of the denominators.
Find the LCM of each set of Polynomials
1) 12y2, 6x2 LCM = 12x2y2
2) 16ab3, 5a2b2, 20ac LCM = 80a2b3c
3) x2 – 2x, x2 - 4 LCM = x(x + 2)(x – 2)
4) x2 – x – 20, x2 + 6x + 8
LCM = (x + 4) (x – 5) (x + 2)
34
23
LCD is 12.
Find equivalentfractions using the LCD.
9
12
812
=9 + 812
=1712
Collect the numerators, keeping the LCD.
Adding Fractions - A Review
Remember: When adding or subtracting fractions, you need a common
denominator!
51
53 . a
54
21
32 . b
63
64
61
43
21
.c34
21
64
32
When Multiplying or Dividing Fractions, you don’t need a common Denominator
1. Factor, if necessary.2. Cancel common factors, if possible.
3. Look at the denominator.
4. Reduce, if possible.5. Leave the denominators in factored form.
Steps for Adding and Subtracting Rational Expressions:
If the denominators are the same, add or subtract the numerators and place the result over
the common denominator.If the denominators are different,
find the LCD. Change the expressions according to the LCD and add or subtract numerators. Place the result
over the common denominator.
Addition and SubtractionIs the denominator the same??
• Example: Simplify
4
6x
156x
2
3x22
52x
33
Simplify...
23x
52x
Find the LCD: 6x
Now, rewrite the expression using the LCD of 6x
Add the fractions...4 15
6x
= 19 6x
65m
8
3m2n
7mn2
15m2n2
18mn2 40n 105m
15m2n2
LCD = 15m2n2
m ≠ 0n ≠ 0
6(3mn2) + 8(5n) - 7(15m)
Multiply by 3mn2
Multiply by 5nMultiply by
15m
Example 1 Simplify:
Examples:
xxa
27
23 .
x24
x2
46
43 .
xxxb
463
xx
4)2(3
xxor
Example 2
3x 23
2x 4x 1
5
15
15x 10 30x 12x 3
15
27x 7
15
LCD = 15
(3x + 2) (5) - (2x)(15) - (4x + 1)(3)
Mult by 5
Mult by 15 Mult by
3
Example 3 Simplify:
2x 14
3x 1
2
5x 33
3(2x 1) 6(3x 1) 4(5x 3)
12
6x 3 18x 6 20x 12
12
8x 21
12
Example 4 Simplify:
4a3b
2b3a
LCD = 3ab
3ab
4a2 2b2
3ab
a ≠ 0b ≠ 0
Example 5
(a) (b)(4a) - (2b)
Simplify:
Adding and Subtracting with polynomials as denominators
Simplify:
3x 6
x 2 x 2
8x 16x 2 x 2
3
(x 2)x 2x 2
8(x 2)
x 2x 2
Simplify...
3x 2
8x 2
Find the LCD:
Rewrite the expression using the LCD of (x + 2)(x – 2)
3x 6 (8x 16)
(x 2)(x 2)
– 5x – 22 (x + 2)(x – 2)
(x + 2)(x – 2)
3x 6 8x 16(x 2)(x 2)
2x 3
3
x 1 (x 3)(x 1)
LCD =(x + 3)(x + 1)
x ≠ -1, -3
2x 2 3x 9(x 3)(x 1)
5x 11(x 3)(x 1)
2 + 3(x + 1) (x + 3)
Multiply by (x + 1)
Multiply by (x + 3)
Adding and Subtracting with Binomial Denominators
233 3634
xxx
x ** Needs a common denominator 1st!
Sometimes it helps to factor the denominators to make it easier to find
your LCD.)12(334
23
xxx
xLCD: 3x3(2x+1)
)12(3)12(3)12(4
3
2
3
xxx
xxx
)12(3)12(4
3
2
xx
xx)12(348
3
2
xxxx
Example 6 Simplify:
91
961
22
xxxx
)3)(3(1
)3)(3(1
xxxx
x
)3()3()3(
)3()3()3)(1(
22
xx
xxxxx
LCD: (x+3)2(x-3)
)3()3()3()3)(1(
2
xxxxx
)3()3(333
2
2
xx
xxxx)3()3(
632
2
xxxx
Example 7 Simplify:
2xx 1
3x
x 2
(x 1)(x 2)
x ≠ 1, -2 2x2 4x 3x2 3x
(x 1)(x 2)
x2 7x
(x 1)(x 2)
2x (x + 2) - 3x (x - 1)
Example 8 Simplify:
3xx2 5x 6
2x
x2 2x 3(x + 3)(x + 2) (x + 3)(x - 1)
LCD(x + 3)(x + 2)(x - 1)
(x 3)(x 2)(x 1)3x
3x2 3x 2x2 4x(x 3)(x 2)(x 1)
x2 7x
(x 3)(x 2)(x 1)x ≠ -3, -2, 1
- 2x(x - 1) (x + 2)
Simplify:Example 9
4xx2 5x 6
5x
x2 4x 4(x - 3)(x - 2) (x - 2)(x - 2)
(x 3)(x 2)(x 2)
4x + 5x
4x2 8x 5x2 15x (x 3)(x 2)(x 2)
9x2 23x (x 3)(x 2)(x 2)
x ≠ 3, 2
(x - 2) (x - 3)
LCD(x - 3)(x - 2)(x - 2)
Example 10 Simplify:
x 3x2 1
x 4
x2 3x 2(x - 1)(x + 1) (x - 2)(x - 1)
(x 1)(x 1)(x 2)
(x + 3) - (x - 4)
(x2 x 6) (x2 3x 4) (x 1)(x 1)(x 2)
4x 2 (x 1)(x 1)(x 2)
x ≠ 1, -1, 2
(x - 2) (x + 1)
LCD(x - 1)(x + 1)(x - 2)
Simplify:Example 11
Pg 173 # 1 – 21 odd