In Exercises 1–30, factor each trinomial, or state that the trino...

Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.

a² + 5a − 14

Accepted Answer

Hello, everyone. We've been asked to perform a factorization in the given expression or categorize it as a prime trinomial. Also, we're going to perform a check using the foil method. The expression we're given is Y squared plus nine, Y minus 36. Our answer choices are a, the parentheses, Y minus nine multiplied by the parentheses. Y plus four B, the parentheses, y -2 multiplied by the parentheses, Y plus 18 C parentheses, Y -3 multiplied by the parentheses. Y plus 12 D, the parentheses, Y plus three multiplied by the parentheses. Y minus 12, rewriting our given expression. Again, I have Y squared plus nine, Y minus 36. Recall that when we factor a trinomial, we're gonna factor it into two binomials. So I'm opening two sets of parentheses next to each other. They are currently empty and we will fill them using the information we get from our trinomial. First thing I'm gonna do is find the first value in both of my binomials and I know that that's the value that I get from the square root in this case of Y squared. So it's gonna be Y because I want my first terms in my binomials to multiply back to the first term in the trinomial and Y times Y will give me Y squared. Next, I need to find two values that not only multiply to negative 36, but add to a positive nine. Recognizing that if it needs to multiply to a negative 36, I'm gonna need a negative number multiplied by a positive number. So I'm gonna put the opposite signs in my parentheses right now. So I don't forget. So one of my parentheses has Y minus the other parentheses has Y plus. Now off to the side, I'm gonna list all the factors I could think of that will multiply to and we'll see if any of them add to a positive nine. So there could be negative one and a positive 36 positive one and a negative 36 -2 and a positive 18, positive two and a negative 18, -3 and a positive 12, positive three and a negative 12, -4 and a positive nine, positive four and a negative nine and negative sex and a positive S six. So now that I've listed all my possible factors, I'm gonna go through and say do any of these add up to a positive nine. If I were to add them together, -1 and positive 36 will not give me a positive nine, neither will one and negative 36. So those are both out -2 and positive 18. That would add to a positive 16. So that's not what we're looking for either. Uh same with positive to a negative 18. That would give me a negative 16. Next -3 and positive 12 that will add to a positive nine. This is my combination. These are the factors I want to use. I'm gonna put the three with the minus sign and the 12 with the plus sign. So at this step, my parentheses are Y -3 multiplied by the parentheses, Y plus 12. This is probably our factorization but we need to check it. So we're gonna check below and recall that the foil method is first outer, inner and last. So that's the order we are multiplying our values together. So first would be the first term in each binomial. So Y multiplied by Y which is Y squared outer is Y multiplied by positive 12, which results in positive 12, Y inner negative three multiplied by Y which is negative three Y and last negative three multiplied by positive 12, which is -36. Currently, I have Y squared plus 12, Y -3, Y -36. I'm going to combine my like terms. So 12 Y and negative three Y and rewrite the whole expression. So Y squared plus nine Y minus this matches my original trinomial. So I know that our factorization is correct. So our solution is the parentheses Y minus three multiplied by the parentheses. Y plus 12. That's answer choice. C have a nice day.

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. a² + 5a − 14

Key Concepts

Factoring Trinomials

Prime Trinomials

FOIL Method

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