Rational Expressions • Much of the terminology and many of the techniques for the arithmetic of fractions of real numbers carry over.
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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 ab
.
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)
x3
(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
.
ba
(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