In Exercises 23–48, factor completely, or state that the polynomi...

Question

In Exercises 23–48, factor completely, or state that the polynomial is prime.x² + 25y²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to completely factories the following polynomial expression if possible and otherwise declare if the expression is a prime polynomial, the polynomial we're given is M cubed plus 36 Q Q two or Q to the third. And we've got for our answer choices, answer choice. A open parentheses. M plus six Q close parentheses multiplied by open parentheses. M squared minus six MQ plus six Q squared. Close parentheses. Answer choice B is open parentheses. M plus six Q closed parentheses multiplied by open parentheses. M squared plus six M Q minus six Q squared. Close parentheses. Answer choice C is open parentheses. M plus six, Q, close parentheses multiplied by open parentheses. M squared plus six, not six Q squared, close parentheses. And answer choice D is that this polynomial is prime, right? The first step in factoring is that we take a look at all of our terms and ask, ask ourselves if they have any factors in common or any variables in common. And this time we have M cubed and 36 Q two the third. And those don't share any common factors or variables. So there's nothing we can factor out in front. And the next thing we do is we take a look and we noticed that a lot of these seem like they're being cubed. And we ask ourselves, could we have, because they're added, could we have a sum of cubes? Everything here would have to be a product of cubes. So M cubed is M times M times M that one checks out and Q cubed or Q to the third is Q times Q times Q, that one checks out. But 36. Well, that's, that doesn't break down to a cubits. 36 would be six times six. And the sixes are two times three and there aren't three of any one factor here. So this is not a sum of cubes and actually, we can't do any factoring here with these cubes, but not all cubes. So this matches with answer choice D, this polynomial is prime. Well done. We'll catch you on the next one.

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In Exercises 23–48, factor completely, or state that the polynomial is prime.x² + 25y²

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