Factor each trinomial completely. See Examples 1 through 7....

Key Concepts

Quadratic Trinomials

A quadratic trinomial is a polynomial expression of the form ax² + bx + c, where a, b, and c are constants. Factoring these expressions means rewriting them as a product of two binomials, provided that such factorization exists in the set of integers.

AC Method

The AC method is a systematic technique for factoring quadratic trinomials with a leading coefficient other than 1. It involves multiplying the coefficient of x² (a) and the constant term (c) to find a product, then identifying two numbers that multiply to this product while summing to the coefficient of x (b).

Factoring by Grouping

After using the AC method to split the middle term, factoring by grouping is used to break the resulting four-term polynomial into two groups. Each group is factored separately, and a common binomial factor is extracted, completing the factorization process.

Finding Factor Pairs

Finding factor pairs involves determining two integers that multiply to a given product (typically a*c in quadratic trinomials) and add up to a specified middle term coefficient (b). This is a critical step in both the AC method and other factoring strategies for quadratic expressions.

Transcript

00:01 We're asked to factor this trinomial completely.

00:03 And the first thing that i always want to do whenever i'm trying to factor is i do want to still look for my gcf.

00:09 It can make the numbers much smaller if it is available.

00:12 However, i noticed the two outside numbers here.

00:15 Those are even and this one is odd.

00:17 And 55 only has 5 and 11.

00:20 And those do not go into that.

00:21 So it does not look like we have a number that goes into all of them.

00:25 I also notice that they do not all have an x, which means there's not an x there.

00:28 So unfortunately, there is no gcf for this problem, which does mean that i'm going to go into two parentheses.

00:37 All right.

00:37 In order to put the numbers in the parentheses, i need to first think about what makes 36, and then i need to think about what makes 14.

00:45 I notice my 14 is negative.

00:47 So because that is negative, i am going to be subtracting over here as i think about my factors, because a positive and a negative will make this negative.

00:55 So that's something good to know ahead of time.

00:57 So the factors of 36 are going to be 1 and 36, 2 and 18, 3 and 12, 4 and 9, and then 6 and 6.

01:10 The factors of 14 are 1 and 14, 2 and 7.

01:15 So thankfully there's not a lot of factors for 14 because there are quite a few for 36 that we need to kind of look at.

01:22 Now remember, you want to multiply these numbers with a set of these numbers, and together, they need to make 55.

01:30 And again, we're going to try to do that by subtracting.

01:32 So what i'm going to do here is i'm just going to kind of use my calculator, see what i can come up with.

01:39 I know 18 times 14 is going to be pretty big at 252.

01:44 So that's probably not going to work...

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