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.15a² − ab − 6b²

Accepted Answer

Hello, today we're gonna be factoring the following trinomial. So what we are given is 20 M squared minus three M N minus 35 N squared. Now, in order to factor this trinomial, we're going to be using the trial and error method. So in order to use this method, we're going to list out the factors of the first term and the last term, the factors of the first term, I'm gonna go ahead and label them as the P factors. And the good thing about the P factors is that all the different combinations of factors are going to be positive. So we need to ask ourselves what are two different numbers or the different combinations of numbers that we can multiply together to give us 20 M squared. Well, we can multiply M and 20 M, we can multiply four M and five M and we can multiply two M and 10 M to give us 20 M squared. Now, what about the factors of the last term? The factors of the last term I'm going to label those as the Q factors. And the tedious thing about the Q factors is that the Q factors can be different combinations of positive and negative integers. So what are two numbers that we can multiply to give us negative 35 N squared. Well, because it's a negative number, we're going to have to multiply a positive and negative together. So the different combinations for negative 35 N squared is going to be positive N and negative 35 N or it could be negative N and positive 35 N. We can also have five N and seven N, I'm sorry, five N and negative seven N or we can have negative five N and positive seven N. And so now what I'm going to do is I'm going to create a two pairs of parentheses in these pairs of parentheses. The first term in each pair is going to be a number from the P factors. And what comes afterward is going to be a number from the Q factors. And both of these can either be a plus or minus combination. So, what we need to do now is pick a different combination of pairs of P factors and Q factors. And we're going to use those to see whether or not we can find the factored form of this trinomial. So let's go ahead and use four M and five M as the P factors. And let's go ahead and try out negative five N and seven N as the Q factors. Now, if we want to go ahead and plug this into, into our parentheses. What this is going to give us is four M minus five N times the quantity five M plus seven N. And a trick to, to verifying whether this is the proper factorization or not is to multiply the very first and the very last term together and multiply the two middle terms together. And once we add up the product of these two pairs of terms, if it sums to give us the original middle term, which in this case is negative 3MN, then this is going to be the correct factorization for the given trinomial. So four M times seven N is going to give us positive 28 M N and negative five N times positive five M is going to give us negative 25 M N and 28 M N minus 25 M N is going to give us positive 35 M N but positive 35 M N isn't the same thing as negative 35 M N. So let's go ahead and pick a different combination of factors. Well, notice that the sum of these products were very close to the middle term. So let's go ahead and use the same pair, but let's go ahead and switch the signs. So instead of having four M minus five N, let's go ahead and use four M plus five N times the quantity five M minus seven N. Now the product of four M and negative seven N is going to give us negative 28 M N and the product of five N with five M is going to give us positive M N. If we sum these two quantities together, what we get is negative three M N, which matches the original middle term for the given trinomial. So that means that the proper factorization for the trinomial is going to be four M plus five N Times five M -7 n. And to verify that this is the answer, we can foil these two quantities together and it should give us the original trinomial. So if we multiply four M with five M, that is going to give us 20 M squared, multiplying four M with negative seven N is going to give us negative 28 M N five N times five M is going to give us positive 25 M N and five N times negative seven N is going to give us negative N squared. And combining the like terms together is going to give us 20 M squared minus three M N minus 35 N squared. And that is going to be the original trinomial in which we are trying to factor. So that verifies that four M plus five N times five M minus seven N is going to be the proper factorization for the trinomial. And with that being said, the answer to this problem is going to be A. So I hope this video helps you in understanding how to factor this trinomial and verify the solution. 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.15a² − ab − 6b²

Key Concepts

Factoring Trinomials

FOIL Method

Prime Trinomials

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