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REVIEW OF ALGEBRA

3

x

x

y

1

2

y

y

x

y

x

x x

y

x

xy

(d)

2

y

x

y

y

x

y

y x

y

xy

y

1

x

x

FACTORING

We have used the Distributive Law to expand certain algebraic expressions. We sometimes

need to reverse this process (again using the Distributive Law) by factoring an expression

as a product of simpler ones. The easiest situation occurs when the expression has a com-

mon factor as follows:

Expanding

3x(x-2)=3x@-6x

Factoring

2

To factor a quadratic of the form

x

bx

c

we note that

2

x

r x

s

x

r

s x

rs

so we need to choose numbers

r and s

so that

r

s

b

and

rs

c

.

2

EXAMPLE 4

Factor

x

5x

24

.

SOLUTION

The two integers that add to give and multiply to give

5

24

are

3

and .

8

Therefore

2

x

5x

24

x

3 x

8

2

EXAMPLE 5

Factor

2x

7x

4

.

2

SOLUTION

Even though the coefﬁcient of

x

is not , we can still look for factors of the

1

form

2x

r

and

x

s

, where

rs

4

. Experimentation reveals that

2

2x

7x

4

2x

1 x

4

Some special quadratics can be factored by using Equations 1 or 2 (from right to left)

or by using the formula for a difference of squares:

2

2

a

b

a

b a

b

3

The analogous formula for a difference of cubes is

3

3

2

2

4

a

b

a

b a

ab

b

which you can verify by expanding the right side. For a sum of cubes we have

3

3

2

2

a

b

a

b a

ab

b

5

EXAMPLE 6

2

2

(a)

x

6x

9

x

3

(Equation 2;

a

x, b

3

)

2

(b)

4x

25

2x

5 2x

5

(Equation 3;

a

2x, b

5

)

3

2

(c)

x

8

x

2 x

2x

4

(Equation 5;

a

x, b

2

)

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Parent category: Education