8

8.5 Add and Subtract Rational Expressions

Complex fraction

A fraction that contains a fraction in its numerator or denominator

ADD (SUBTRACT) WITH LIKE DENOMINATORS

To add (or subtract) rational expressions with like denominators, simply add (or subtract) their numerators . Then place the result over the common denominator. Let a, b, and c be polynomials with c ¹ 0

Addition Subtraction:

Example 1

Add with like denominators

Add numerators

ADD (SUBTRACT) WITH UNLIKE DENOMINATORS

To add (or subtract) two rational expressions with unlike denominators, find the
least common denominator (LCD), which is the _least common multiple__ (LCM) of the denominators.

Rewrite each rational expression using the LCD, then add (or subtract) using the procedure for like denominators. Let a, b, c, and d be polynomials with c ¹ 0 and d ¹ 0.

Addition

Subtraction

Example 2

Add with unlike denominators

Add:

To find the LCD, factor each denominator and write the highest power to which each factor occurs. Note that 4x² = 2x²x² and 2x² + 4x = _2x(x + 4)____ , so the LCD

is __2²x²(x + 4)_______ = __4x²(x + 4)__.

= +

Example 3

Subtract with unlike denominators

Subtract:

(x – 3)(x +3)

3(x – 3)

=

3(x – 3)

7

(x + 3)

(x + 3)

(x – 3)(x +3)











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SIMPLIFYING COMPLEX FRACTIONS

A complex fraction is a fraction that contains a fraction in its numerator or denominator .

Method 1: If necessary, simplify the numerator and denominator by writing each as a single fraction. Then divide the numerator by the denominator.

Method 2: Multiply the numerator and the denominator by the least common denominator (LCD) of every fraction in the numerator and denominator. Then simplify.

Example 4

Simplify a complex fraction (Method 1)