Transcript Rational Expressions • Much of the terminology and many of the techniques for the arithmetic of fractions of real numbers carry over.

Rational Expressions
• Much of the terminology and many of the techniques for the
arithmetic of fractions of real numbers carry over to algebraic
fractions, which are the quotients of algebraic expressions.
• In particular, the quotient of two polynomials is referred to as a
rational expression.
• The rules for multiplying and dividing rational expressions are the
same as those for multiplying and dividing fractions of real
numbers. Do you recall what they are?
• To simplify a rational expression, use the cancellation principle:
ab b
 , a  0.
ac c
Factoring and Cancellation
• Simplification of a rational expression is often a two-step
process: (1) Factor, and (2) Cancel.
x2  4
.
• Problem. Simplify 2
x  5x  6
Solution. (1) Factor numerator and denominator:
x2  4
( x  2)(x  2)

,
2
x  5 x  6 ( x  3)(x  2)
(2) Cancel common factors:
( x  2)(x  2) ( x  2)

.
( x  3)(x  2) ( x  3)
• Warning!!! Only multiplicative factors can be cancelled.
Adding and Subtracting Rational Expressions
• When two rational expressions have the same denominator, the
addition and subtraction rules are:
a b ab
 
.
c c
c
• To add or subtract rational expressions with different
denominators, we must first rewrite each rational expression as
an equivalent one with the same denominator as the others.
1 1
• Example. Add the rational expressions:   ?
2 3
Since the denominators are different, we convert each
expression to an equivalent expression with denominator 6.
1 1 3 2 5
    .
2 3 6 6 6
Least Common Denominator
• Although any common denominator will do for adding rational
expressions, we will concentrate on finding the least common
denominator, or LCD, of two or more rational expressions.
• The LCD is found by a 3-step process: (1) Factor the
denominator of each fraction, (2) Find the highest power
(final factor) to which each factor occurs, and (3) The LCD is
the product of the final factors.
x 8
3
, 2
.
• Example. Find the LCD: 4
2
x  4x x  2x
(1) x 4  4 x 2  x 2 ( x  2)(x  2), x 2  2 x  x( x  2)
(2) Final Factorsare x 2 , ( x  2), ( x  2)
(3) LCD is x 2 ( x  2)(x  2).
Addition of Rational Expressions
• Addition of rational expressions is a 3-step process:
(1) Find the LCD.
(2) Write each expression as an equivalent expression which
has denominator equal to the LCD.
(3) Add the rational expressions from Step 2.
• Example.
x 8
3

?
2
2
x  4 x  2x
(1) LCD  x( x  2)(x  2)
x 8
3
x( x  8)
3( x  2)
(2)



( x  2)(x  2) x( x  2) x( x  2)(x  2) x( x  2)(x  2)
x 2  5x  6
( x  2)(x  3)
x3
(3) T hesum is :


x( x  2)(x  2) x( x  2)(x  2) x( x  2)
Complex Fractions
• We want to simplify a complex fraction, which is a fractional
form with fractions in the numerator or denominator or both.
• Simplifying a complex fraction is a 2-step process.
(1) Find the LCD of all fractions in the numerator and
denominator.
(2) Multiply both numerator and denominator by the LCD.
• Example.
1
b
.
1 1

a b
(1) The LCD is ab.
result is
a
.
ba
(2) The
Summary of Rational Expressions; We discussed:
• Rational expressions
• The cancellation principle
• Addition of rational expressions with the same denominator
• Least common denominator (LCD) and how to find it
• Addition of general rational expressions as a 3-step process
• Simplification of complex fractions as a 2-step process