Factoring

Let's first review some product rules (where we have binomials we are FOILing ):
• Monmial times binomial 2x*(3x + 6) = 2x*3x + 2x*6 = 6x² + 12x We multiplied 2x times BOTH terms in the ( )
• Binomial times binomial both have involve subtraction: (x - 3)(x - 2) = x² -2x -3x + 6 = x² -5x + 6
• Binomial times binomial both involve addition: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6
• Binomial times binomial one subtraction, one addition, plus bigger: (x + 3)(x - 2) = x² -2x + 3x -6 = x² + x - 6
• Binomial times binomial one subtraction, one addition, minus bigger: (x - 3)(x + 2) = x² +2x - 3x -6 = x² - x - 6

Now let's briefly look at the results. If we multiplied two binomials

and both involved addition all of the terms in the resulting trinomial were positive.

(x + 3)(x + 2) = x² + 5x + 6

and both involved subtraction the middle term of the resulting trinomial was negative but the last term was positive (minus times minus is positive)

(x - 3)(x - 2) = x² -5x + 6

and one involved addition and one subtraction the last term in our trinomial was negative and the middle term was positive or negative depending on the size of the terms involved.

(x - 3)(x + 2) = x² - x - 6

(x + 3)(x - 2) = x² + x - 6

We can use the above information we try to factor a trinomial. That is, when we want to go from a trinomial back to the two binomials that were multiplied to get our trinomial. (Note: The following only deals with the case where the exponent of the squared term is a 1).

1. The first term is squared. If we multiplied two binomials and got x² we must have multiplied x times x so we can write: (x ± ?)(x ± ?)
2. If the middle AND last term are both positive we know that we must have (x + ?)(x + ?), Both signs are positive
3. If the middle term is negative and the last term is positive we know that both signs are negative, (x - ?)(x - ?)
4. If the middle and last term are both negative we know that the signs are opposites (x + ?)(x - ?)
5. If the last term is negative we know the signs are opposites, the sign of the middle term tells us whether the plus or minus term is the larger.
6. We know that the two numbers were multiplied to give us the last term and added to get the middle term: (x ± a)(x ± b) = x² ± ax ± bx ± ab.

Some examples:
• x² + 8x + 7, middle and last positive, (x + 7)(x + 1), last number was a 7, it is likely that it comes from 1*7, middle term is 8x from 1x + 7x
• x² - 5x - 14, middle and last terms are minus, signs must alternate, likely 14 comes from 1*14, or 2*7. If signs alternate 7-2 = 5 works for middle term, (x - 7)(x + 2) is the answer
• x² -6x + 8, middle term is minus but last term is +, must have like signs that are negative, 8 comes from 1*8 or 2*4, 2*4 works, (x - 4)(x - 2)
• x² + 7x - 8, middle term is +, last is -, signs must be opposite, 8 like to come from 1*8 or 2*4, 1*8 works if we have (x + 8)(x - 1)

Shown below are some problems for you to try.