In Exercises 69–82, factor completely.

12x²y – 46xy² + 14y³

Question

In Exercises 69–82, factor completely.

12x²y – 46xy² + 14y³

Accepted Answer

Hello, today we're gonna be factoring the following trinomial. So what we are given is 120 X squared Y minus 102 X Y squared plus 18 Y cubed. Now, before we factor the trinomial, let's go ahead and see if we can factor out a greatest common factor. So looking at the co coefficients, we have 121 102 and and these three coefficients have a common multiple of six. So we can go ahead and factor out a six from the given trinomial. Furthermore, each term has at least one Y inside of it. So we can also factor out A Y from the given trinomial. So if we factor out six Y from the original trinomial, what we are left with is six Y times the quantity X squared minus 17 X Y plus three Y squared. And now what we're going to do is we're going to factor this using the trial and error method. So the trial and error method states that we list out all the possible terms or all the possible factors of the first term which are going to be the P factors and these factors are all going to be positive. So what are the possible factors for 20 X squared? Well, that's going to be X and 20 X or it can be five X and four X or it could be two X and 10 X. And we want to go ahead and list out the factors of the last term which are going to be the Q factors and the Q factors are all the possible factors of the combination of positive and negative numbers. So the possible factors for three Y squared can be three Y times Y or it can be negative three Y times negative Y. So now what I'm going to do is I'm going to create two groups of parentheses. And what's gonna go inside of these parentheses is going to be the leading terms which are the numbers from the P factors. And the last terms which are going to be the numbers from the Q factors and the Q factors can be either positive or negative. So let's go ahead and pick any combination of P and Q factors to test out. So let's go ahead and test five X and four X and let's go ahead and test negative three Y and negative Y. So putting this into our parentheses is going to give us five X minus three Y Times four X -Y. And in order to verify that this is the correct factorization, we're gonna multiply the first term with the very last term and we're gonna multiply the two middle terms together. And if the sum of these products add up to give us - XY, then this is going to be the correct factorization. So five X times negative Y is going to give us negative five X Y and negative three Y times four X is going to give us negative 12 X Y and negative five X Y minus 12 X Y is going to give us N 17 X Y which matches the middle term. So five X minus three Y times four X minus Y is the correct factorization of the trinomial. But we want to be careful because keep in mind we factor out a six Y in the beginning of the problem. So if we combine all of our factors together, we get six Y times the quantity five X minus three Y times the quantity four X minus Y. And this is going to be the completely factored version of the original trinomial. And with that being said, the answer to this problem is going to be C 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.

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In Exercises 69–82, factor completely. 12x²y – 46xy² + 14y³