Factor each polynomial completely. See Example 6.4m²p - 12mnp + 9...

Question

Factor each polynomial completely. See Example 6.4m²p - 12mnp + 9n²p

Accepted Answer

Welcome back. I am so glad you're here. We're asked to rewrite the following expression by factoring it completely. Our expression is nine X squared, Z minus 30 XYZ plus 25 Y squared Z. Our answer choices are answer choice A three Z multiplied by the quantity of X minus five Y squared. Answer choice BZ multiplied by the quantity of three X minus five Y squared. Answer choice C three Y multiplied by the quantity of X minus five Z squared and answer choice dy multiplied by the quantity of three X minus five Z squared. All right. The first step when we've got so much going on here, we want to look at all three of our terms to see if they have any factors in common that we can factor out. So our first term nine X squared Z, our second term negative 30 XYZ and our third term positive 25 Y squared Z, we can start by comparing the coefficients. We've got a nine, a 30 it's a negative 30. But let's just look at it as a 30 for a moment and then a if we factor these out nine would be three multiplied by three. If we break down 30 that is three multiplied by 10, which is two multiplied by five. So three multiplied by two, multiplied by five and 25 that's five multiplied by five. So there isn't actually a factor that all three of those share. But it's interesting the pattern here, nine and 25 are both squares. And then the 30 shares a three in common with the nines and A five in common with the 20 fives. And then it just has that too. That'll be more interesting in a little bit. But anyway, nothing we can factor out of that. Now, let's look at the variables. The first term has an X squared in A Z. The second term has an X as well and, and A Z as well. Also A Y and the third term has no XS, but it does also have a Z. So all three terms have one single Z that we can factor out in front. We put a Z in front and then that's multiplied by inside of parentheses. Now we'll just take all three of those terms and divide them by Z nine X squared Z multiplied or excuse me, divided by Z is just nine X squared, negative 30 XYZ divided by Z is negative XY and 25 Y squared Z divided by Z is just plus 25 Y squared. So we've got that Z factored out. But looking at what's left, this looks like it has the potential to be a perfect square trinomial. Here's how we know if we have a perfect square trio meal in previous lessons, we recall from previous lessons, we recall that a perfect square trinomial it might be adding in the middle or it might be subtracting this one subtracting. So we would have an A squared minus two A B plus B squared. Our first term nine X squared, those are both that the nine and the X squared are both squares. So what would A be, well, to get the nine, we're squaring three. So A would be three and then to get the X squared, we're squaring X. So A would be three X if we look at the third term 25 Y squared, the 25 and the Y squared are both squares. If we want to find B, what are we squaring to get 25? That's a five. What are we scoring to get Y squared? That's A Y. So we have an A squared and we have A B squared. We know what our A and our B are now in the middle. We're saying we think that that's negative two multiplied by A which is three X multiplied by B which is five Y. Well, we already demonstrated up here that if we take three multiplied by two, multiplied by five and then one of them is negative, we would get that negative 30 and then what's left is just the X and the Y. So yes, this is a perfect square trinomial which is wonderful because we can factor a perfect square trim, no meal pretty quickly. A perfect square trinomial factors to be open parentheses. A minus B squared. So we set our expression equal two. We can't forget our Z. Now, that's multiplied by the quantity of our A which is three X minus our B which is five Y closed parentheses squared. And that's it. Now, we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Factor each polynomial completely. See Example 6.4m²p - 12mnp + 9n²p

Key Concepts

Factoring Polynomials

Common Factors

Quadratic Trinomials

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