Page 144 - MarceAlgebra Demystified

CHAPTER 6 Factoring 131

Reducing a fraction or adding two fractions sometimes only requires that 1
y x
be factored from one or more denominators. For instance in the
x y
numerator and denominator are only off by a factor of 1. To reduce this
fraction, factor 1 from the numerator or denominator:
y x ð y þ xÞ ðx yÞ 1
¼ ¼ ¼ ¼ 1or
x y x y x y 1
y x y x y x 1
¼ ¼ ¼ ¼ 1:
x y ð x þ yÞ ðy xÞ 1

3 x
In the sum þ the denominators are off by a factor of 1. Factor
y x x y a a
1 from one of the denominators and use the fact that ¼ to write
b b
both terms with the same denominator.
3 x 3 x 3 x
þ ¼ þ ¼ þ
y x x y ð y þ xÞ x y ðx yÞ x y
3 x 3 þ x
¼ þ ¼
x y x y x y
In the next examples and practice problems a ‘‘ 1’’ is factored from the
denominator and moved to the numerator.

Examples

1 1 1 1 3 3 3
¼ ¼ ¼ ¼ ¼
1 x ð 1 þ xÞ ðx 1Þ x 1 x 6 ðx þ 6Þ x þ 6

3x 3x ð 3xÞ 3x
¼ ¼ ¼
14 þ 9x ð 14 9xÞ 14 9x 14 9x
16x 5 16x 5 16x 5 ð16x 5Þ 16x þ 5
¼ ¼ ¼ ¼
7x 3 ð 7x þ 3Þ ð3 7xÞ 3 7x 3 7x

Practice

1
1: ¼
y x