Warm up. Adding and Subtracting Rational Expressions Goal 1 Determine the LCM of polynomials Goal 2...
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Transcript of Warm up. Adding and Subtracting Rational Expressions Goal 1 Determine the LCM of polynomials Goal 2...
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)
3
4
2
3
LCD is 12.
Find equivalentfractions using the LCD.
9
12
8
12
=9 + 8
12
=17
12
Collect the numerators, keeping the LCD.
Adding Fractions - A Review
Remember: When adding or subtracting fractions, you need a common
denominator!
5
1
5
3 . a
5
4
2
1
3
2 . b
6
3
6
4
6
1
43
21
.c3
4
2
1
6
4
3
2
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
15
6x
2
3x
2
2
5
2x
3
3
Simplify...
2
3x
5
2xFind the LCD: 6x
Now, rewrite the expression using the LCD of 6x
Add the fractions...4 15
6x
= 19 6x
6
5m
8
3m2n
7
mn2
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:
3x 2
3 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:
4a
3b
2b
3a
LCD = 3ab
3ab
4a2 2b2
3ab
a ≠ 0b ≠ 0
Example 5
(a) (b)(4a) - (2b)
Simplify:
ab
ba
3
)2(2 22
Adding and Subtracting with polynomials as denominators
Simplify:
3x 6
x 2 x 2
8x 16
x 2 x 2
3
(x 2)
x 2
x 2
8
(x 2)
x 2
x 2
Simplify...
3
x 2
8
x 2Find 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)
2
x 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
233 363
4
xx
x
x ** Needs a common denominator 1st!
Sometimes it helps to factor the denominators to make it easier to find
your LCD.)12(33
423
xx
x
xLCD: 3x3(2x+1)
)12(3)12(3
)12(43
2
3
xx
x
xx
x
)12(3
)12(43
2
xx
xx)12(3
483
2
xx
xx
Example 6 Simplify:
9
1
96
122
xxx
x
)3)(3(
1
)3)(3(
1
xxxx
x
)3()3(
)3(
)3()3(
)3)(1(22
xx
x
xx
xx
LCD: (x+3)2(x-3)
)3()3(
)3()3)(1(2
xx
xxx
)3()3(
3332
2
xx
xxxx
)3()3(
632
2
xx
xx
Example 7 Simplify:
2x
x 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:
)2)(1(
)7(
xx
xx
3x
x2 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
)1)(2)(3(
)7(
xxx
xx
4x
x2 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:
)2)(2)(3(
)239(
xxx
xx
x 3
x2 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
)2)(1)(1(
)12(2
xxx
x