Factoring – Signs in Trinomials

• When we factor trinomials into binomials, it is very important to understand the possible sign combinations.

• The L in the FOIL pattern for multiplying two binomials stands for “Last”.

• Example 1

( 3)( 5)x x 2 2 15x x

Last Last Last× =

2 2 15x x

The last term of the trinomial is negative.

This means that the last terms of the binomials must be opposite is sign. This can be done in two ways.

• Turn this process around, starting with the trinomial, and determine the signs for the binomial.

• Example 2

or

2 2 15x x

2 2 15x x

2 2 15x x

It doesn’t make any difference at this point which one you write. Adjustments can be made later in the factoring process.

2 6 8x x

The last term of the trinomial is positive.

This means that the last terms of the binomials must be the same sign.

• Example 3

2 6 8x x

Since the middle term is negative, use the two negatives.

2 6 8x x

2 6 8x x

The last term of the trinomial is positive.

This means that the last terms of the binomials must be the same sign.

• Example 4

2 6 8x x

Since the middle term is positive, use the two positives.

2 6 8x x

SUMMARY

Trinomial

1. Third term negative

Binomials

Opposite signs

2. Third term positive Same signs

a) Middle term positive

Both signs positive

b) Middle term negative

Both signs negative

2 7 12x x

Since the last term of the trinomial is positive, the signs must be the same.

Since the middle term is negative, both signs are negative.

• Example 5

2 7 12x x

2 12x x

Since the last term of the trinomial is negative, the signs must be opposite.

• Example 6

2 12x x