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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?. Fractions require you to find the Least Common Multiple ( LCM) in order to add and subtract them! .

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slide1

Adding and Subtracting Rational Expressions

Goal 1 Determine the LCM of polynomials

Goal 2 Add and Subtract Rational Expressions

slide2

What is the Least Common Multiple?

Fractions require you to find the Least Common Multiple (LCM) in order to add and subtract them!

  • Least Common Multiple (LCM) - smallestnumber or polynomial into which each of the numbers or polynomials will divide evenly.

The Least Common Denominator is the LCM of the denominators.

slide3

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)

slide4

Adding Fractions - A Review

LCD is 12.

Find equivalent

fractions using the

LCD.

Collect the numerators,

keeping the LCD.

slide5

Remember: When adding or subtracting fractions, you need a common denominator!

When Multiplying or Dividing Fractions, you don’t need a common Denominator

slide6

Steps for Adding and Subtracting Rational Expressions:

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.

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.

slide7

Addition and Subtraction

  • Is the denominator the same??
  • Example: Simplify

Find the LCD: 6x

Now, rewrite the expression using the LCD of 6x

Simplify...

Add the fractions...

=19

6x

slide8

Simplify:

Example 1

+ 8(5n)

- 7(15m)

6(3mn2)

Multiply by

5n

LCD = 15m2n2

m ≠ 0

n ≠ 0

Multiply by

15m

Multiply by

3mn2

slide9

Examples:

Example 2

slide10

Simplify:

Example 3

(3x + 2)

- (2x)

(15)

- (4x + 1)

(3)

(5)

LCD = 15

Mult by

15

Mult by

3

Mult by

5

slide11

Simplify:

Example 4

slide12

Simplify:

Example 5

- (2b)

a ≠ 0

b ≠ 0

(4a)

(a)

(b)

LCD = 3ab

slide13

Adding and Subtracting with polynomials as denominators

Simplify:

Find the LCD:

(x + 2)(x – 2)

Rewrite the expression using the LCD of (x + 2)(x – 2)

Simplify...

– 5x – 22

(x + 2)(x – 2)

slide14

Adding and Subtracting with Binomial Denominators

(x + 3)

(x + 1)

2

+ 3

Multiply by

(x + 3)

Multiply by

(x + 1)

LCD =

(x + 3)(x + 1)

x ≠ -1, -3

slide15

Simplify:

Example 6

** Needs a common denominator 1st! Sometimes it helps to factor the denominators to make it easier to find your LCD.

LCD: 3x3(2x+1)

slide16

Simplify:

Example 7

LCD: (x+3)2(x-3)

slide17

Simplify:

Example 8

(x + 2)

- 3x

(x - 1)

2x

x ≠ 1, -2

slide18

Simplify:

Example 9

3x

(x - 1)

- 2x

(x + 2)

(x + 3)(x + 2)

(x + 3)(x - 1)

LCD

(x + 3)(x + 2)(x - 1)

x ≠ -3, -2, 1

slide19

Simplify:

Example 10

4x

+ 5x

(x - 3)

(x - 2)

(x - 3)(x - 2)

(x - 2)(x - 2)

LCD

(x - 3)(x - 2)(x - 2)

x ≠ 3, 2

slide20

Simplify:

Example 11

(x - 2)

(x + 1)

(x + 3)

- (x - 4)

(x - 1)(x + 1)

(x - 2)(x - 1)

LCD

(x - 1)(x + 1)(x - 2)

x ≠ 1, -1, 2