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How to simplify rational expressions, find the least common denominator, add fractions, and subtract them. Learn the steps for adding and subtracting rational expressions. Examples provided for better comprehension.
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Do Now Simplify: ?2 4 ?2+ 6? + 9 ?2+ ? 6 1
Adding and Subtracting Rational Expressions Goal 1 Determine the LCD of polynomials Goal 2 Add and Subtract Rational Expressions
What is the Least Common Denominator? Fractions require you to find the Least Common Multiple (LCM) in order to add and subtract them!
Adding Fractions - A Review LCD is 12. 12+8 =9+8 12 9 3 4+2 = 12 3 Find equivalent fractions using the LCD. =17 Collect the numerators, keeping the LCD. 12
Remember: When adding or subtracting fractions, you need a common denominator! 4 3 4 3 1 2 1 1 = = + = . . a b 6 6 5 5 5 3 2 6 When Multiplying or Dividing Fractions, you don t need a common Denominator 1 4 2 1 4 2 = = = . c 3 6 2 3 3 4
Steps for Adding and Subtracting Rational Expressions: 1. Factor, if necessary. 2. Cancel common factors, if possible. 3. Look at the denominator. 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. 4. Reduce, if possible. 5. Leave the denominators in factored form.
Addition and Subtraction Is the denominator the same?? Example 1a: Simplify 2 3x+5 Find the LCD: 6x 2x Now, rewrite the expression using the LCD of 6x =2 2 2 +5 3 3 Simplify... 3x 2x 4 6x+15 =4 +15 6x = 6x Add the fractions... = 19 6x
Examples: Example 1 + ( 3 ) 2 x + 3 x 6 3 6 x x or + = . b 4 x 4 4 x 4 x
Adding and Subtracting with polynomials as denominators Simplify: x +2 3 8 Find the LCD: (x + 2)(x 2) x 2 Rewrite the expression using the LCD of (x + 2)(x 2) x 2 x 2 x + 2 x + 2 3 8 = Simplify... (x + 2) (x 2) 3x 6 x + 2 8x +16 x + 2 = ) ( ) x 2 ( ( ) x 2 ( ) =3x 6 8x 16 (x + 2)(x 2) =3x 6 (8x +16) (x + 2)(x 2) 5x 22 (x + 2)(x 2)
Adding and Subtracting with Binomial Denominators You Try!!! 2 3 (x + 3) (x + 1) 2 + 3 = =(x + +3)(x + +1) = =2x + +2+ + 3x + +9 (x + +3)(x+ +1)= = Multiply by (x + 1) (x + 3) x+ + 3+ + x+ +1 5x + +11 (x + +3)(x + +1) LCD = (x + 3)(x + 1) Multiply by x -1, -3
Simplify: + 1 1 x + 1 x 1 x = + + 2 2 6 9 9 x x x + + + ) 3 ( 3 )( ) 3 ( 3 )( x x x LCD: (x+3)2(x-3) + = ) 3 + + ) 3 x ) 3 + ( x 1 )( 2 ( ( 1 )( x ) 3 2 ( x x x x x x = + ) 3 ) 3 + ) 3 ) 3 + ) 3 2 ( ) 3 ( ( ( x x ( ( x + 2 3 x 3 3 x x x x x 2 3 2 6 x x = = ) 3 + ) 3 2 ( ( x ) 3 + ) 3 ( ( x
You Try!!!! Simplify: = = (x 1)(x+ + 2) = =2x2+ + 4x 3x2+ + 3x (x 1)(x + + 2) x2+ + 7x (x 1)(x + + 2) 2x x 1 3x x + +2 2x(x + 2) - 3x (x - 1) x 1, -2 = =
Simplify: 3x 2x 3x (x - 1) - 2x (x + 2) = =(x + +3)(x + +2)(x 1) x2+ +5x + +6 (x + 3)(x + 2) x2+ +2x 3 (x + 3)(x - 1) = =3x2 3x 2x2 4x (x + + 3)(x + + 2)(x 1) x2 7x (x + + 3)(x + + 2)(x 1) LCD (x + 3)(x + 2)(x - 1) x -3, -2, 1 = =
You try!!!! Simplify: 4x 5x 4x + 5x (x - 3) (x - 2) x2 5x + +6+ + (x - 3)(x - 2) = =(x 3)(x 2)(x 2) x2 4x + +4 (x - 2)(x - 2) 4x2 8x + + 5x2 15x (x 3)(x 2)(x 2) 9x2 23x (x 3)(x 2)(x 2) LCD (x - 3)(x - 2)(x - 2) = = x 3, 2 = =
Simplify: x + +3 x2 1 (x - 1)(x + 1) x 4 (x - 2) (x + 1) (x + 3) - (x - 4) = =(x 1)(x + +1)(x 2) x2 3x + +2 (x - 2)(x - 1) = =(x2+ + x 6) (x2 3x 4) (x 1)(x + +1)(x 2) LCD (x - 1)(x + 1)(x - 2) 4x 2 x 1, -1, 2 = = (x 1)(x + +1)(x 2)
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