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.
Example 1 Factor 5x^{ 2}  17x + 14 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. Since the constant term is positive we know the signswill be the same. Since we want the factors to add up to 17x thesigns will both have to be negative. Keep this in mind. 1 st : Since the coefficient of the first term is prime (5),we will start with the first term. Find the factors of the firstterm. The factors of 5x^{ 2} are 1x and 5x. These go inthe first positions. We can also go ahead and put in the signs(both negative)
2 nd : Find the factors of the constant term. The factors of14 are 1,14 and 2, 7. Remember, we need the inside/outsidecombination to add up to the middle term which is 17x. This timewe don't just consider the factors of the constant term becausethe first term also had factors. Here's where the guessing comesin. Let's try the factors 2,7 and see what happens. (1x  2 ) (5 x  7 ) Let's check the inside/outsidecombination. If we multiply inside, 2 times 5x gives us 10x.Multiplying outside 1x times 7 gives us 7x. Add up theinside/outside combination: 10x + 7x is equal to 17x which isour middle term. We made a lucky guess!! Note: If the 2 and 7hadn't worked, we should try 7 and 2. If that didn't work either,we would try 1 and 14 or 14 and 1. The point is, you keep tryinguntil you find the right combination. Check by using FOIL (x  2) (5x  7)
Example 2 Factor 8x^{ 2}  10x + 3 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. Since the constant term is positive we know the signswill be the same. Since we want the factors to add up to 10x thesigns will both have to be negative. Keep this in mind. 1 st : Since the coefficient of the constant term is prime(3), we will start with the constant term. Find the factors ofthe first term. The factors of 3 are 1 and 3. These go in thelast positions. We can also go ahead and put in the signs (bothnegative) 2 nd : Find the factors of the first term. The factors of 8x^{2} are 1x, 8x and 2x, 4x. Remember, we need theinside/outside combination to add up to the middle term which is10x. 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 4x ,2x and seewhat happens. (4x  1 ) (2x  3 ) Let's check the inside/outsidecombination. If we multiply inside, 1 times 2x gives us 2x.Multiplying outside 4x times 3 gives us 12x. Add up theinside/outside combination: 2x + 12x is equal to 14x which isNOT our middle term. This is not a lucky guess. Ok, let's reversethe factors 4x, 2x and try it that way: (2x  1 ) ( 4x  3) Let's check the inside/outsidecombination. If we multiply inside, 1 times 4x gives us 4x.Multiplying outside 2x times 3 gives us 6x. Add up theinside/outside combination: 4x + 6x is equal to 10x which isour middle term. Check by using FOIL (2x  1) (4x  3)
