After studying this lesson, you will be able to:
Steps of Factoring:
1. Factor out the GCF
2. Look at the number of terms:
3. Factor Completely
4. Check by Multiplying
This lesson will concentrate on the second step of factoring:Factoring Trinomials.
**When there are 3 terms, we are factoring trinomials. Don'tforget to look for a GCF first.**
Factoring trinomials often requires some trial and error.Don't get frustrated. Try all possible combinations. In theprevious problems, the first term has not had a coefficient. Wewill now look at problems that do have coefficients in the firstterm. This adds another level of trial and error or"guessing".
One thing that will make the "guessing" moreaccurate is to look for a prime number in the first term or theconstant term. Remember, a prime number only has 2 factors.....1and itself. If the coefficient of the first term or the constantterm is prime, start there and "lock in" those factors.
Factor 6x 2 - 13x - 5
This is a trinomial (has 3 terms). There is no GCF other thanone. So, we start with 2 parentheses:
Using our signs rules, we can determine the signs for thefactors. The constant term is negative so we know the signs willbe different. Keep this in mind.
1 st : Since the coefficient of the constant term is prime(5), we will start with the constant term. Find the factors ofthe constant term. The factors of 5 are 1 and 5 . These go in thelast positions. We won't put the signs in yet because we aren'tsure where they go.
2 nd : Find the factors of the first term. The factors of 6x2 are 1x, 6x and 2x, 3x. Remember, we need theinside/outside combination to add up to the middle term which is-13x. This time we don't just consider the factors of theconstant term because the first term also had factors. Here'swhere the guessing comes in. Let's try the factors 2x,3x and seewhat happens.
Notice we still didn't put in the signs. Let's checkthe inside/outside combination. If we multiply inside, 1 times 3xgives us 3x. Multiplying outside 2x times 5 gives us 10x. Nowremember, we have to have different signs. On the inside/outsidecombination we have 3x and 10x. Using different signs, we cannotmake the combination equal the middle term. We resort toguessing. Let's reverse the 2x and 3x and see what happens.
Notice we still didn't put in the signs. Let's checkthe inside/outside combination. If we multiply inside, 1 times 2xgives us 2x. Multiplying outside 3x times 5 gives us 15x. Nowremember, we have to have different signs. On the inside/outsidecombination we have 2x and 15x. If we make the 15x negative andthe 2x positive we will have the combination of -13x...which isour middle term.
(3x + 1 ) (2x - 5)
Check by using FOIL (3x + 1) (2x - 5)