In Exercises 69–82, factor completely.

8a²b + 34ab – 84b

Question

In Exercises 69–82, factor completely.

8a²b + 34ab – 84b

Accepted Answer

Hello, today we're gonna be factoring the following trinomial. So what we are given is six N M squared plus 34 N M -56 n. Now, before we factor this using the trial and error method, let's go ahead and check to see if we could factor out a greatest common factor from the given trinomial. So if we take a look at the coefficients, we have 6:34 and 56, which all have a common multiple of two. So we can go ahead and factor out a two from the current trinomial. Also notice that there is at least one N in each term of the given trinomial. So we can go ahead and factor out an N from the given trinomial. So the current trinomial has the greatest common factor of two N. And if we factor out two N from the trinomial, what we are left with is three M squared plus 17 M minus 28. And now we can go ahead and use the trial and error method on the current trinomial. So the trial and error method states that we want to write out the factors of the first term. And the last term, the first term factors are going to be called the P factors, which can only be factors of positive numbers. And for the last term, the last term are going to have the Q factors which can have a combination of positive and negative numbers. So let's go ahead and write out the terms or the factors for the first term. So the first term has a coefficient of three which is a prime number, which means that the only pair of factors that are possible for the first term are going to be M times three M. And for the last term, since we can use any combination of positive and negative numbers and with 28 being a negative number, the possible factors are going to be one and negative negative one and positive 28. We can also do two and negative 14, negative two and positive 14. And we have a combination of negative four and seven and positive four and negative seven. So now what I'm going to do is I'm going to write two pairs of parentheses. And what's going to go in the first terms for each pair parentheses is going to be a number from the P factors. And the last term is going to be a number from the Q factors and Q can be plus or minus. So the factor form of the current trinomial is going to be P plus or minus Q. So now what we wanna do is we want to pick a combination of factors to test and we're going to be testing M and three M And let's go ahead and test out positive four and -7. So if we plug this in to our current form of P plus or minus Q, what we end up with is three M plus four Times M -7. And now in order to determine whether this is the correct factorization or not, we're gonna multiply the first term with the very last term and the two middle terms together. And we're going to take the sum of those two products. And if the sum of the two products add up to give us this middle term 17 M, that is going to be the correct factorization for the given trinomial. So three M times negative seven is going to give us negative M and four times M is going to give us positive four M and 21 M plus four M is going to give us negative 17 M but that doesn't match up with positive 17 M but very close to it. So this is not the correct factorization, but since our sum came very similar to the middle term, why don't we go ahead and switch the signs for both of the groups? So let's go ahead and test out three M minus four times M plus seven and continuing the same pattern. We're going to multiply the first and very last term together and we're gonna multiply the two middle terms together three M times seven is going to give us positive 21 M and negative four times M is going to give us negative four M and 21 M minus four M is going to give us positive 17 M which matches the middle term in our trinomial. So that means that the factored form of three M squared plus 17 M minus 28 is going to be three M minus four times M plus seven. But we want to be careful because keep in mind we factored out A two N in the beginning. So we wanna go ahead and combine all our factors together and doing so is going to give us two N times the quantity three M minus four times M plus seven. And this is going to be the factored form of the original trinomial. And with that being said, the answer to this problem is going to be B 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 69–82, factor completely. 8a²b + 34ab – 84b

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Factoring by Grouping

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