In mathematics, factorization or factoring is the decomposition of an object (for example, a number or a polynomial) into a product of other objects, or factors, which when multiplied together give the

original.

For example, the number 15 factors into primes as 3 × 5

In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something

to "basic building blocks”

ftrees

To factor an expression means to:

re-write it as a product

Why factor?

Once an expression has been factored, equivalent factors can divide to one

Remember factor trees?

9

3 3X

9 1X

9 = 3 X 3

Notice:

4(x + 2) = 4x + 8

Expanding

Factoring

Common Factoring

A common factor can be divided out of every term in the polynomial

1. Number: GCF2. Variable: Common Variable, take

the lowest exponent

Common Factor:

10x + 20 = 10( )

25x2 + 100x = 25x( )

4x + 12x2 – 16x3 =

4x(1 + 3x – 4x2)

8a2b3 + 12a4b2 =

4a2b2(2b + 3a2)

10x + 2

25 xx + 4

YOU MAY DIVIDE OUT COMMON FACTORS

YOU MAY NOT “CROSS OUT” OR “CANCEL” SIMILAR TERMS

Consider the following example:

3x + 63

, x = 2Direct

= 3(2) + 63

= 123

= 4

Factoring3x + 6

3= 3(x + 2)

3

1

1

= x + 2

= 4

Crossing out

3x + 63

= x + 6

= 8

Stop here…

Trinomials: ax2 + bx + c

Simple (a = 1):

Add to the middle

Multiply to the last

Factorx2 + 5x + 6 = (x )(x ) + 2 + 3

simple

Add: 5 Multiply: 6

Factorx2 - 2x - 15 = (x )(x ) - 5 + 3

simple

Add: -2 Multiply: -15

Complex: (a > 1) Decomposition

Add to the middle, multiply to (first)(last)

Common Factor twice

Factor

6x2 – 1x – 2 = 6x2 – 4x + 3x – 2

= 2x(3x – 2) + 1(3x – 2)

= (3x – 2)(2x + 1)

You may also guess and check

complexAdd: -1 Multiply: -12CF the first pair, then CF the second pair

Difference of Squares

x2 – y2 = (x – y)(x + y)

x2 – 25 = (x – 5)(x + 5)

4x2 – 36y2= (2x – 6y)(2x + 6y)

Grouping

Sometimes, part of a 4 term polynomial can be “grouped” together and factored

Factor

ax + cx + ay + cy

= x(a + c) + y(a + c)

= (a + c)(x + y)

Factor

a2 – p2 + 2a + 1

= a2 + 2a + 1 – p2

=(a + 1)2– p2

=(a + 1 - p)(a + 1 + p)

The Great Factoring Plan!!!

C.F. 3+

3

2

D of SSimple

Complex

Grouping

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