Factor each trinomial, if possible. See Examples 3 and 4.

Question

Factor each trinomial, if possible. See Examples 3 and 4. $24 a^{4} + 10 a^{3} b - 4 a^{2} b^{2}$

Accepted Answer

Welcome back. I am so glad you're here. We are given this polynomial 81 X to the fourth plus 63 X to the third Y minus 30 X squared Y squared. And we are asked to factor it. Alright. We were called from previous lessons that we are going to look at these three terms. And we are going to ask ourselves, is there anything that we can factor out? Are there any factors that they share? So specifically, we'll look at the parts of the terms that could share factors. So 81 63 and negative 30, these whole numbers here. Did they share any common factors? Yes, they are all divisible by three because they're all divisible by the greatest common factor three, we can factor out a three. Now, we can look at these X terms here. We've got X to the fourth X to the third and X squared. And we recall from previous lessons that we are going to look for the X term with the lowest degree, which in this case is this X squared. So everything is divisible by X squared. Or in other words, we can factor out an X squared. There aren't wise in all of our three terms here. The first one doesn't have A Y so we can't factor out A Y. So this is the most we can factor out. Now what's left on the inside of the parentheses while we can divide X to the fourth by three X squared, that 81 divided by three leaves us with and that X to the fourth divided by X squared. Well, we recall from previous lessons that we're going to subtract the exponents when we're dividing variables with exponents. So four minus two, that's two. So we've got X squared left. All right. Now we're taking 63 X to the third Y divided by three X squared. 63 divided by three is 21. So plus 21 X to the third divided by X squared, three minus two is one. So we've got X to the first or just X and then we can't forget to bring down that. Why? Alright. And this last one negative 30 X squared Y squared divided by three X squared. We're factoring out the three X squared, negative 30 divided by three equals negative 10. So minus 10 X squared divided by X squared. That's one. We don't have to put a times one there because negative 10 times one is negative 10 and then bringing down our Y squared. Alright, we factored out that three X squared and now we can look at what's left in the parentheses and ask ourselves if we can factor this further. So what we're doing is kind of the reverse of the foil process. So we look at this first term and that would be the result of the firsts being multiplied inside of our parentheses. And so what times what could give us 27 X squared? Well, we're going to have an X times an X to get that X squared. But what about getting that 27 we've got options for 27 27 could be a nine times a three. It could also be a 27 times a one. Order doesn't matter for the first one. So it could also be three times nine or one times 27. But to have fewer options to test out, we'll just keep it as nine times three and 27 times one because order doesn't matter for the first one. Now, the next thing we would look at is what will get us these lasts and that's going to multiply and become this negative 10 Y squared. So what times what gives us negative 10 Y squared? Well, we'll have the Y times the Y and now we just need to think about this negative 10 while order does matter for the second one. And for negative 10, we've got a lot of different options we could have. It will have to be a positive times a negative. So we could have a negative one times a positive 10, we could also have a positive one times a negative 10. And because order does matter, we could also have a negative 10 times a positive one or a positive times a negative one. But that's not all, we could also have a negative five times a positive two or a positive five times a negative two or we could have a negative two times a positive five or a positive two times a negative five. And why is this important? Well, what also has to happen is we have to choose the factors that not only multiply to get to our firsts in our last two B 27 X squared and negative Y squared. They also have to add up to be a positive 21 X Y. That's when we take our outsides plus our insides. And that would give us our positive 21 X Y. So we know that we've got it narrowed down our first will multiply to get to our first. Our last will multiply to get to our last. Now, we've got to figure out how to get our outsides and insides to add up to 21 X Y. And the way we do that is we will take a factor from our X columns. So we can take a nine Time's a factor from Ry columns. We can take nine times negative one And then add that to the corresponding factor from our X column. So we've got to keep the pairs together. So that would be a three Times the corresponding factor from Ry column that remains three times 10. So keeping these pairs together because then we know they'll still multiply to equal our first and our last. So let's see if this adds up to a positive 21. So we have nine times negative one, that's negative nine plus three times 10 is negative nine plus 30 is positive 21. That is luck. We, if that one didn't work, we'd have to go through and try the nine and three with the pair of one in negative 10 and so on and so forth. But the first one worked this time, which is great. And so now we need to put them in the appropriate spaces in our parentheses. So for the first, we have the nine x And we have the three x there and now we know that we took the nine times negative one. Those were our outsides. So the negative one has to pair with the nine, the negative one will have to go here on the end. So that's part of the outsides and the three times the positive 10 will be our insides. So we'll have a plus 10 Y here and those will be our insides. All right, this is as factored as it gets. We don't need the one in front of the Y for this second set of parentheses here that can just be written as three X minus Y. We look at our answer choices and three X squared times nine X nine, X plus 10, Y times three X minus Y is answer choice. D well done. We'll catch you on the next one.

Factor each trinomial, if possible. See Examples 3 and 4. $24 a^{4} + 10 a^{3} b - 4 a^{2} b^{2}$

Key Concepts

Factoring Trinomials

Greatest Common Factor (GCF)

Factoring by Grouping

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Factor each trinomial, if possible. See Examples 3 and 4. 24a4+10a3b−4a2b224a^4+10a^3b-4a^2b^2

Key Concepts

Factoring Trinomials

Greatest Common Factor (GCF)

Factoring by Grouping

Watch next

Factor each trinomial, if possible. See Examples 3 and 4. $24 a^{4} + 10 a^{3} b - 4 a^{2} b^{2}$