Algebra Demystified 2nd Ed

148 algebra De mystif ieD

PRACTICE

Use factoring by grouping to factor the expression.

1. 6xy^2 + 4xy + 9xy + 6x =


2. x^3 + x^2 − x − 1 =


3. 15xy + 5x + 6y + 2 =


4. 2x^4 − 6x − x^3 y + 3y =


5. 9x^3 + 18x^2 − x −2 =

✔SOLUTIONS

1. 6xy^2 + 4xy + 9xy + 6x = 2xy(3y + 2) + 3x(3y + 2)
   = (2xy + 3x)(3y + 2) = x(2y + 3)(3y + 2)


2. x^3 + x^2 − x − 1 = x^2 (x + 1) − 1(x + 1) = (x^2 − 1)(x + 1)


3. 15xy + 5x + 6y + 2 = 5x(3y + 1) + 2(3y + 1)
   = (5x + 2)(3y + 1)


4. 2x^4 − 6x − x^3 y + 3y = 2x(x^3 − 3) − y(x^3 − 3) = (2x − y)(x^3 − 3)


5. 9x^3 + 18x^2 − x − 2 = 9x^2 (x + 2) − 1(x + 2) = (9x^2 − 1)(x + 2)

Factoring to Simplify Fractions

Factoring algebraic expressions serves many purposes in mathematics, and one of

the more important purposes is in working with fractions. For now, we will sim-

plify fractions by factoring the numerator and denominator and dividing out com-

mon factors. Later, we will use what we learned to perform fraction arithmetic.

EXAMPLES

Factor the numerator and denominator, and then write the fraction in

lowest terms.

62

410

2

2

xxy

xy xy

+

−

Each term in the numerator and denominator has a factor of 2x.

62

410

2

2

xxy

xy xy

+

−

=^23

22 5

3

25

xxy

xxyy

xy

xy y

()

()

+

−

= +

−

xy x

xxy

xy

xxy

y

xy

−

−

−

−

= −

16 −

1

16

1

(^216)
()
()
= 
PRACTICE
Use factoring by grouping to factor the expression.
EXAMPLES
Factor the numerator and denominator, and then write the fraction in