In Exercises 45–68, use the method of your choice to factor each ...

Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.

15x² + 111xy − 14y²

Accepted Answer

Hello, today we're gonna be factoring the following trinomial and verifying its solution by using the foil method. So what we are given is 13 A squared plus 75 A B minus 20 B squared. Now, in order to factor this trinomial, we're going to be using the trial and error method. The trial and error method tells us to list out the factors of the first term and the last term. Now the factors of the first term which are going to be the P factors are any combination of factors of only positive numbers. So that means that the factors for 13 a squared is only going to be 13 a times a because the coefficient of 13 is a prime number. Now, what about the factors for the last term? Well, the factors for the last term are going to be the Q factors and the Q factors could be any combination of factors of positive and negative numbers. So for negative 20 B squared, we can multiply negative B with positive 20 B or we can do positive B with negative 20 B, we can also try five B and negative four B or negative five B with positive four B. And we can also try two B with negative 10 B Or -2 B with positive 10 b. Now that we have all the factors listed, what I'm going to do is create two pairs of parentheses. And in these pairs of parentheses, the first terms are going to be the numbers from the P factors. And the last term is going to be the fact, the numbers from the Q factors and the Q can be positive or negative. So now what we're going to do is we're going to pick any combination of P N Q factors and test them to see what the factor form is going to be. So let's go ahead and try out 13 a and negative B and 20 b. We're gonna plug this in and this is going to give us 13 A minus B times the quantity A plus 20 B. Now, in order to verify whether this is the factored form or not, let's go ahead and first multiply the very first term with the very last term and the two middle two terms together. So 13 A times 20 B is going to give us 260 A B and negative B times A is going to give us negative A B and that is going to add up to give us 259 A B. But in order to check whether this is the correct factorization or not, this sum has to add up to the middle term, which in this case is positive 75 A B, which is not the same thing as 259 A B. So this is not going to be the factorization. And if we try the next factors, which is just us switching the signs in the current test, this is just going to switch the sign of the current sum which is going to give us negative 259 A B which is still not going to be the correct factorization. So the B and 20 B factors don't work. Now, let's go ahead and test the five B and four B factors. So that's going to give us 13 A plus five B times A minus four B, 13 A Times -4 B Is going to give us -52 A B and five B times A is going to give us positive five A B. Now 52 A B, I'm sorry, negative 52 A B plus five B is going to give us negative 47 A B and that doesn't sum up to negative 75 A B. And if we were to test the opposite signs that would just change the sign of the current sum, which is going to give us positive 47 A B. So the five B and four B factors don't work. Now, we just have one more pair to test and that's going to be the two B and 10 B factors. So if we test that out, we get 13 a plus two B times A minus 10 B, 13 A times negative 10 B is going to give us negative 130 A B and two B times A is going to give us positive two A B and 100 and 30 A B plus two A B is going to give us negative 128 A B which does not sum up to positive 75 A B. And if we test the opposite signs, it's just gonna reverse the sign of the sum, which is going to give us 128 A B. So that means that the factors of two B and 10 B don't work. So after testing out all the factors, none of the factors for the Q and P method work, which means that the current trinomial is in its prime form, it is not factor. Thus, the solution to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 15x² + 111xy − 14y²

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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