Introduction to factoring polynomials

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if the sign is minus: use different signs, one plus and one minus

“smile” to make sure you get the middle term

multiply the inner most terms together then multiply the outer most terms together, and add the two products together.

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Factor completely: 2x2 – 5x – 7

Factors of the first term: 1x & 2x

Factors of the last term: -1 & 7 or 1 & -7

(2x – 7)(x + 1)

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Factor Completely. 4x2 + 83x + 60

Nothing common

Factors of the first term: 1 & 4 or 2 & 2

Factors of the last term: 1,6 2,30 3,20 4,15 5,12 6,10

Since each pair of factors of the last has an even number, we can not use 2 & 2 from the first term

(4x + 3)(1x + 20 )

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Sign Pattern for the Binomials

Trinomial Sign Pattern Binomial Sign Pattern

+ + ( + )( + )

- + ( - )( - )

- - 1 plus and 1 minus

+ - 1 plus and 1 minus

But as you can tell from the previous example, the FOIL method of factoring requires a lot of trial and error (and hence luck!) . . . Better way?

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Your Turn to Try a Few

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ac Method for factoring ax2 + bx + c

Factor out the GCF, if any

For the remaining trinomial, multiply ac

Look for factors of ac that sum to b

Rewrite the bx term as a sum using the factors found in step 3

Factor by grouping

Check by multiplying using FOIL

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Ex: Factor 3x2 – 4x – 15

1. Is there a GCF?

No

2. Multiply ac 

a =

3

3

and c =

– 15

– 15

3(-15) = - 45

3. Factors of -45 that sum to

– 4

– 4

4. Rewrite the middle term

3x2 – 4x – 15 = 3x2 – 9x + 5x – 15

1 – 45  – 44

3 – 15  – 12

5 – 9  – 4

Note: although there are more factors of – 45, we don’t have to check them since we found what we were looking for!

Four-term polynomial . . . How should we proceed to factor?

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Factor by grouping . . . 3x2 – 9x + 5x – 15

Common factor among the 1st two terms?

 3 x 2 – 9x = 3x( – )

3x

3x

3x

3

x

3

Common factor among the 2nd two terms?

5

 5 x – 15 = 5( – )

5

5

3

x

3

 3x2 – 9x + 5x – 15 = 3x(x – 3) + 5(x – 3)

= (x – 3)( )

3x

+ 5

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