**Factoring Strategies**

**Greatest Common Factor**

Distribution Property

ab + ac = a (b + c)

ab - ac = a (b - c)

**Three Terms**

Standard form: ax + bx + c

**Two Terms**

Difference of Squares

a – b = ( a + b)(a - b)

Sum of Squares

a + b = Prime

Sum and Difference of Cubes

a + b = ( a + b )( a - ab + b )

a - b = ( a - b )( a + ab + b )

For the signs of the factors

Same Opposite Always Positive

**Four Terms**

Factor by Grouping

1) Factor GCF of all terms.

2) Factor GCF of pairs of terms.

(the negative GCF used when - in middle)

3) Factor Common Binomial Factors.

Berry Method

x +bx + c

(x + p)(x + q)

a = 1

AC Method

"Solve the Riddle

Split the middle"

Solve the Riddle

Write the Factors

Take out the GCF's

Solve the riddle:

Factors of c

Sum is b

Answer: p and q

Example

24x y + 30x y

(6x y )(4x ) + (6x y )(5y)

6x y (4x + 5y)

4 5 2 6

2

2

2 2

2 2

Count the number of terms

3 3 2 2

3 3 2 2

Just remember SOAP

What are Factors?

Multiply Factors to get a Product

One, Two,

Three,

Four Terms

Ha Ha Ha

2 5 2 2 5

2 5 2

GCF : 6x y

2 5

Rewrite each term using the GCF

Use the Distributive Property to factor out the GCF

1) Multiply ac.

2) Find 2 integers whose product is ac and the sum is b:

3) Rewrite the middle term: {pq = ac and p+q=b}

4) Factor by Grouping. {bx = px + qx}

ax + bx + c

2

Find Two numbers that are: Factors of AC and Sum is B

FACTOR

Write the factors

Sign Chart:

If C > 0 then the factors are the same sign

they will both be the sign of B.

If C < 0 then the factors are different signs

the larger factor is the same sign as B

Example

AC = -12 factors different signs

B = -1 larger factor (-)

Solve the Riddle: -4 and +3

2x - x + 6

(2x - 4)(2x - 3)

(x - 2)(2x - 3)

2

Write the two factors using ax with p and q:

Factor out the GCF 2 from first factor and discard

What is left is the answer

Examples

x + 7x + 12

(x + 3)(x + 4)

x - 8x + 12

(x - 6)(x - 2)

x + 7x - 18

(x + 9)(x - 2)

x - 6x - 16

(x - 8)(x + 2)

2

2

2

2

C = +12 factors same sign

B = +7 factors both (+)

Solve the Riddle: +4 and +3

C = +12 factors same sign

B = -8 factors both (-)

Solve the Riddle: -6 and -2

C = -18 factors different signs

B = +7 larger factors (+)

Solve the Riddle: +9 and -2

C = -16 factors different signs

B = -6 larger factors (-)

Solve the Riddle: -8 and +2

Examples

2x + 7x + 6

2x + 4x + 3x + 6

2x(x + 2) + 3(x + 2)

(2x + 3)(x + 2)

3x + 7x - 6

3x + 9x - 2x - 6

3x(x + 3) - 2(x +3)

(3x - 2)(x + 3)

2x - 5x + 3

2x - 3x - 2x + 3

x (2x - 3) -1 (2x - 3)

(2x - 3)(x - 1)

AC = +12 factors same sign

B = +7 both factors (+)

Solve the Riddle: +4 and +3

AC = -18 factors different signs

B = +7 larger factors (+)

Solve the Riddle: +9 and -2

AC = +6 factors same signs

B = -5 both factors (-)

Solve the Riddle: -3 and -2

Example

25x - 9

(5x) - (3) 49y + 16

(5x + 3)(5x - 3) PRIME

2

2

2

Difference of Squares

Sum of Squares

2

Step #1 Factor out the GCF

2

2

2

2

2

2

Finding the GCF

GCF is the product of the common factors

Always Choose the smallest exponent

24x y ; 30x y

4 5 2 6

24x y = 2 3 x y

30x y = 2 3 5 x y

4 5 3 4 5

*

2 6 2 6

* *

Factor each completly

GCF is 6x y

2 5

Factor Negative

Leading coefficient is Negative

-12x + 18 -2x + 8x - 14x

(-6)(2x) + (-6)(-3) -2x(x ) + -2x(-4x) + -2x(7)

-6(2x - 3) -2x(x - 4x + 7)

2

3

2

2

When your GCF is negative

the remaining factor is an opposite:

all the signs change

Step #2

Factor a Negative GCF

-ab + ac = -a (b - c)

Step #1

Example

3xy - 6x + 5y - 10

3x(y -2) + 5(y - 2)

(y - 2)(3x + 5)

8x - 20xy - 6xy + 15y

4x(2x - 5y) - 3y(2x - 5y)

(2x - 5y)(4x - 3y)

Conclusion

Steps to Success

Difference of Squares

3. Factor using AC Method

Grouping

1. Factor out the GCF

2. Count the Number of Terms

Step #3 Choose a method based on the number of terms you have