Download Steps for Factoring

A Summary of Rational Expressions
To Determine When a Rational Expression Is Undefined:
1. Set the denominator equal to 0.
2. Factor and solve for the variable.
x2  2
undefined?
2 x 2  13 x  15
Set 2x 2  13 x  15  0 and then solve
(2x  3)( x  5)  0
Ex. When is
2x  3  0 or x  5  0
2x  3
x5
3
2
2
x 2
3
So,
is undefined when x  or x  5
2
2 x  13x  15
2
x
To Simplify a Rational Expression:
1. Factor both the numerator and the denominator.
2. Cancel common factors.
16 x 2  24 x  9
4x  3
First factor both the numerator and the denominator
Ex. Simplify
(4 x  3)(4 x  3)
 4x  3
4x  3
Multiplying Rational Expressions:
1. Factor numerators and denominators.
2. Cancel common factors.
3. Multiply numerators and multiply denominators.
4. Recheck that the expression is in simplest form.
2 x2  x  6 2 x2  5x  3

2 x 2  3 x  2 2 x 2  11x  12
First factor all of the expressions, and then cancel.
Ex. Multiply
(2 x  3)( x  2) (2 x  1)( x  3) ( x  3)


(2 x  1)( x  2) (2 x  3)( x  4) ( x  4)
Dividing Rational Expressions:
1.
Change division to multiplication; multiply the first fraction by the reciprocal of the
second fraction.
2.
Follow the steps for multiplying rational expressions.
x 2  y 2 3x  3 y

9
27 x 2
First rewrite the problem as a multiplication problem.
Ex. Divide:
x 2  y 2 27 x 2

9
3x  3 y
Now complete using the procedures for multiplication.
3
( x  y)( x  y) 27 x 2

 x2 ( x  y)
9
3( x  y)
Adding/Subtracting Rational Expressions:
1.
Add or subtract numerators and write over the LCD.
2.
Factor numerator and denominator if possible.
3.
Cancel common factors.
Ex. Add:
x2  2x  8  x2  x  5

x 2  8 x  15 x 2  8 x  15
Add the numerators and place over the LCD.
x2  2 x  8  x2  x  5
x3
 2
2
x  8 x  15
x  8 x  15
Finally, factor the numerator and denominator and then cancel common factors.
( x  3)
1

( x  3)( x  5) ( x  5)
Ex. Subtract:
x2  4 x  8 2 x2  4 x 1

x 2  5 x  24 x 2  5 x  24
Distribute the negative sign and place over the LCD. Then combine like terms.
x2  4 x  8  2 x2  4 x  1
 x2  9

x 2  5 x  24
x 2  5 x  24
Finally, factor the numerator and denominator and then cancel common factors.
1( x 2  9)
1( x  3)( x  3) 1( x  3)


( x  3)( x  8)
( x  3)( x  8)
( x  8)