Introduction to Factoring Trinomials - reverse of FOIL

YayMath.org  "Factoring means turn it into pieces you can multiply." 
►Accompanying worksheet for this video.



A trinomial has 3 terms.   x² + 2x - 63
A binomial has 2 terms.   x - 7
The 2 factors of a trinomial are binomials, and each can be written in parentheses.
(x + 9)(x - 7)  They are called binomial factors.
When these binomial factors are multiplied, they will equal the trinomial.

After checking for a greatest common factor and factoring it out, then you can factor the trinomial.

To easily determine signs when factoring trinomials:

1.  If the sign of the last term in a trinomial is negative, such as x² + 2x - 63
the signs between the terms in the binomial factors will be one positive and one negative.
Example 1:  
x² + 2x - 63
(x + 9)(x - 7)

Example 2:
x² - 2x - 63
(x - 9)(x + 7)

Do you see the difference?
• In the first example, only the last term in the trinomial was negative.
• In the second example, both the last term and the middle term were negative.
• So no matter what the middle term is, if the last term in a trinomial is negative, the signs between the terms in the binomial factors will be one positive and one negative.

So what does the sign of the middle term in the trinomial tell us?
It is what you use to determine which of the binomial factors will be positive, and which will be negative.
• In the first example, the middle term of the trinomial is positive (+2x), showing that the 9 in the binomial factors should be positive since it is greater than 7, because if you combine +9 and -7, you will get +2.
• In the second example, the middle term of the trinomial is negative (-2x), showing that the 9 in the binomial factors should be negative, because if you combine -9 and +7, you will get -2.


2.  If the sign of the last term of a trinomial is positive, the signs between the terms of the binomial factors will either be BOTH positive or BOTH negative.
• If the last term in the trinomial is + 63, the terms in the binomial factors will either be 
( __ - 7) and ( __ - 9), or they will be ( __ + 7 and ( __ + 9).
• Then the sign of the middle term of the trinomial will determine what they both will be.

Example 3:
x² + 2xy + y²
(x + y)(x + y)

Example 4:
x² - 2xy + y²
(x - y)(x - y)

If you use the FOIL method and multiply the Inner and Outer terms (from the binomial factors), you will get either both positive terms or both negative terms to add together, equaling the middle term of the trinomial.
In example 3, +xy and +xy will give you +2xy.
In example 4, -xy and -xy will give you -2xy.


►Found this video at Virtual Nerd: Determining Signs when Factoring a Trinomial