rational expressions much of the terminology and many of the techniques for the arithmetic of...
TRANSCRIPT
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:
.0 , ac
b
ac
ab
Factoring and Cancellation
• Simplification of a rational expression is often a two-step process: (1) Factor, and (2) Cancel.
• Problem. Simplify
Solution. (1) Factor numerator and denominator:
(2) Cancel common factors:
• Warning!!! Only multiplicative factors can be cancelled.
.65
42
2
xx
x
,)2)(3(
)2)(2(
65
42
2
xx
xx
xx
x
.)3(
)2(
)2)(3(
)2)(2(
x
x
xx
xx
Adding and Subtracting Rational Expressions
• When two rational expressions have the same denominator, the addition and subtraction rules are:
• 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.
• Example. Add the rational expressions:
Since the denominators are different, we convert each expression to an equivalent expression with denominator 6.
.c
ba
c
b
c
a
?3
1
2
1
.6
5
6
2
6
3
3
1
2
1
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.
• Example. Find the LCD:.
2
3,
4
8224 xxxx
x
2).2)(( is LCD (3)
2)( 2),( , are Factors Final )2(
)2(2 ),2)(2(4 )1(
2
2
2224
xxx
xxx
xxxxxxxxx
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. ?2
3
4
822
xxx
x
)2(
3
)2)(2(
)3)(2(
)2)(2(
65 :is sum The )3(
)2)(2(
)2(3
)2)(2(
)8(
)2(
3
)2)(2(
8 )2(
)2)(2(LCD (1)
2
xx
x
xxx
xx
xxx
xx
xxx
x
xxx
xx
xxxx
x
xxx
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) The LCD is ab.
(2) The result is
.11
1
ba
b
.ab
a
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