### Canceling Like Factors

When we reduce a common fraction such as

we do so by noticing that there is a factor common to both the numerator and the denominator (a factor of 2 in this example), which we can divide out of both the numerator and the denominator.

We use exactly the same procedure to reduce rational expressions.

#### Polynomial / Monomial

Each term in the numerator must have a factor that cancels a common factor in the denominator.

,

but

cannot be reduced because the 2 is not a common factor of the entire numerator.

WARNING  You can only cancel a factor of the entire numerator with a factor of the entire denominator

 However, as an alternative, a fraction with more than one term in the numerator can be split up into separate fractions with each term over the same denominator; then each separate fraction can bereduced if possible:               · Think of this as the reverse of adding fractions over a common denominator. Sometimes this is a useful thing to do, depending on the circumstances.You end up with simpler fractions, but the price you pay is that you have more fractions than you started with.

·  Polynomials must be factored first. You can’t cancel factors unless you can see the factors:

Example:

· Notice how canceling the (x – 2) from the denominator left behind a factor of 1

### Multiplication and Division

Same rules as for rational numbers!

#### Multiplication

• Both the numerators and the denominators multiply together
• Common factors may be cancelled before multiplying

Example:

 Given Equation: First factor all the expressions: (I also put the denominators in parentheses because then it is easier t  see them as distinct factors) Now cancel common factors—any factor on the top can cancel with any factor on the bottom: Now just multiply what’s left. You usually do not have to multiply outthe factors, just leave them as shown.

#### Division

• Multiply by the reciprocal of the divisor
• Invert the second fraction, then proceed with multiplication as above
• Do not attempt to cancel factors before it is written as a multiplication