Factoring Trinomials
Case 1: lead coefficient is one.
In this case we need to find a pair of numbers m and n whose product is the constant term (c), and whose sum is the coefficient of x (b). The m and n then become the constant terms in our binomial factors.
Try:
Case 2: Lead coefficient not equal to one, Trial method.
In this case we observe that the lead term of the trinomial must be the product of the lead terms in our binomial factors. Furthermore the constant term must be the product of the constants in our binomial factors. If we look at all possible factors of the lead term and constant term we can, by trial and error, determine which combination will form the middle term when foiling.
Try:
Example: Factor
We must find an m and n such that
and
and will work so…
Test with foil:
First:
Outside:
Inside:
Last:
Then,
As needed
(Notice how the -3x and 5x add to the 2x and how the 5 and the -3 multiply to -15 when foiling)
Example: Factor
The possible factorings of are:
and
The possible factorings of 6 are:
and
(It is often necessary to allow your factors to be negative)
(Ex :)
Then list all possible factoring that contain these pieces and find the “oi” part of foil to determine witch one give you the 11x you need.
Thus,
(In practice you can stop once you find it)
Case 3: Lead coefficient not equal to one, “ac” method.
The trial method works well when the lead term and constant term have a small number of factors. Lots of factors can mean lots of cases to check. The “ac” method removes the need for these long trials in exchange for a more technical methodology.
To factor first find the product of the lead coefficient (a) and the constant term (c). The, like in case one, find a pair of numbers p and q that multiply to make this product and add to the coefficient of the x term (b). Then replace the x term with the sum
And factor your new four term polynomial by grouping.
Try:
Example: Factor
First find the product (ac)
We must find an p and q such that
and
and will work so replace the 2x with or just to get…
then factor
by grouping
so,
(Finding p and q is not always easy. Just list all the ways to factor (ac) and add up each pair. Be careful of your signs. If none of the pairs add up to the right number then the trinomial is simply not factorable, (or you left off a pair).)