Introduction to factoring polynomials

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a3 + b3 or a3 – b3

Is the trinomial a perfect-square trinomial?

a2 + 2ab + b2 or a2 – 2ab + b2

Is the trinomial a product of two binomials?

Factored completely?

Slide 32

FOIL Method of Factoring

Recall FOIL

(3x + 4)(4x + 5) = 12x2 + 15x + 16x + 20 = 12x2 + 31x + 20

The product of the two binomials is a trinomial

The constant term is the product of the L terms

The coefficient of x, b, is the sum of the O & I products

The coefficient of x2, a, is the product of the F terms

Slide 33

FOIL Method of Factoring

Factor out the GCF, if any

For the remaining trinomial, find the F terms ( x + )( x + ) = ax2

Find the L terms ( x + )( x + ) = c

Look for the outer and inner products to sum to bx

Check the factorization by using FOIL to multiply

Slide 34

Ex: Factor b2 + 6b + 5

1. there is no GCF

2. the lead coefficient is 1  (1b )(1b )

3. Look for factors of 5

1, 5 & 5, 1

(b + 1)(b + 5) or (b + 5)(b + 1)

4. outer-inner product?

(b + 1)(b + 5)  5b + b = 6b

or (b + 5)(b + 1)  b + 5b = 6b

Either one works  b2 + 6b + 5 = (b + 1)(b + 5)

5. check: (b + 1)(b + 5) =

b2 + 5b + b + 5

= b2 + 6b + 5

Slide 35

Ex: Factor y2 + 6y – 55

1. there is no GCF

2. the lead coefficient is 1  (1y )(1y )

3. Look for factors of – 55

1, -55 & 5, - 11 & 11, - 5 & 55, - 1

(y + 1)(y – 55) or (y + 5)(y - 11) or ( y + 11)(y – 5) or (y + 55)(y – 1)

4. outer-inner product?

(y + 1)(y - 55)  -55y + y = - 54y

(y + 55)(y - 1)  -y + 55y = 54y

 y2 + 6y - 55 = (y + 11)(y – 5)

5. check: (y + 11)(y – 5) =

y2 – 5y + 11y - 55

= y2 + 6y – 55

(y + 5)(y - 11)  -11y + 5y = -6y

(y + 11)(y - 5)  -5y + 11y = 6y

Slide 36

Factor completely – 3 Terms

Always look for a common factor

immediately take it out to the front of the expression all common factors

show what’s left inside ONE set of parenthesis

Identify the number of terms.

If there are three terms, and the leading coefficient is positive:

find all the factors of the first term, find all the factors of the last term

Within 2 sets of parentheses,

place the factors from the first term in the front of the parentheses

place the factors from the last term in the back of the parentheses

NEVER put common factors together in one parenthesis.

check the last sign,

if the sign is plus: use the SAME signs, the sign of the 2nd term