Download MTH 098 - Shelton State

MTH 092
Section 12.1
Simplifying Rational Expressions
Section 12.2
Multiplying and Dividing Rational Expressions
Fractions Again?!?!?!?
• A rational expression is of the form
P
Q
Where P and Q are polynomials with Q not
equal to 0.
Why Can’t Q be equal to 0?
• Recall, from our work with slope, that division
by 0 is undefined.
• If Q is a polynomial, then it has variables.
• Those variables cannot take on values that
cause Q to become 0.
• To figure out what those values are, set Q = 0
and solve.
• Key words: undefined, domain
Examples
• Find any numbers for which each rational
expression is undefined:
3
5x
x 1
5x  2
19 x 3  2
x2  x
11x 2  1
x 2  5 x  14
Reducing To Lowest Terms
• Recall that reducing a fraction means dividing
the numerator and denominator by the same
value (usually the greatest common factor).
• Reducing a rational expression involves two
steps:
1. Factor both the numerator and denominator.
2. Cancel common factors.
• Cancel factors, not terms.
Examples: Reduce to Lowest Terms
y9
9 y
x5
x 2  25
9 x  99
x 2  11x
x 4  10 x 3
2
x  17 x  70
4x2  4x 1
2
2x  9x  5
Multiplying and Dividing
1. Factor the numerators and denominators
completely.
2. Cancel common factors.
3. Remember that when you are dividing, you
must multiply by the reciprocal of the second
rational expression.
4. You can not cancel terms. You can not cancel
parts of terms.
Multiply or Divide
4 x  24 5

20 x x  6
2
x  3  5 x  15
5
25
x 2  9 x  20 x 2  11x  28
 2
2
x  15 x  44 x  12 x  35
x 1
20 x  100

2
2 x  5x  3
2x  3
Opposite Factors Make -1
• For all real numbers a and b,
a b
 1
ba
• The -1 is usually put in the numerator and can
be distributed.
Apply “the rule of -1”
• Reduce, multiply, or divide as indicated:
y 9
9 y
49  y 2
y7
9 x 5 3b  3a

2
2
a b
27 x 2
3y
12 xy
 2
3 x x 9