In Exercises 95–104, factor completely.

x⁶ − 9x³ + 8

Question

In Exercises 95–104, factor completely.

x⁶ − 9x³ + 8

Accepted Answer

Welcome back. I am so glad you're here. We are asked to perform complete factorization of the following polynomial expression. The expression we're given is K raised to the sixth power minus 28 K raised to the third power or K cubed plus 27. And our answer choices are answer choice. A, this polynomial is prime answer. Choice B is open parentheses. K minus one closed parentheses multiplied by open parentheses. K minus three closed parentheses multiplied by open parentheses. K squared plus K plus one closed parentheses multiplied by open parentheses. K squared plus three, K plus nine closed parentheses. Answer choice C is open parentheses. K minus one closed parentheses, cubed multiplied by open parentheses, K minus three closed parentheses, cubed and answer choice D is open parentheses, K minus one closed parentheses multiplied by open parentheses. K minus three closed parentheses, multiplied by open parentheses. K squared plus one closed parentheses, multiplied by open parentheses. K squared plus three closed parentheses. All right. The first step in factoring any polynomial is we look at all of our terms and ask ourselves if there's anything we can factor out in front. Our first term here is A K raised to the sixth power. It's only coefficient is one and the last term is a constant term. It doesn't have a variable. So there aren't any common factors that we can factor out. So the next thing we'll do because we have three terms is we will set up two open sets of parentheses and see if we can do the reverse of the foil process. We are going to ask ourselves what multiplied by what equals K raised to the sixth power. Those will be our firsts. And we've got options for different things that multiply to equal K race to the sixth power. But typically for factoring, we want our firsts to be the same if possible. And so here we can have K raised to the third power or K cubed multiplied by K cubed. And so now we have some firsts established. We ask ourselves what multiplied by what equals a positive 27. Those will be our lasts. But they also, these numbers also need two when multiplied, multiplying our outsides and adding those to our insides need to add up to be a negative 28 k cubed. So again, what multiplied by what equals positive 27 but adds up to a negative 28 that's going to be a minus 27 a minus one. All right, we've got this factored and we look at both of these factors and we recognize that we're not done yet because both of these are a difference of cubes. So for this first one, we'll take a look at the first one K cubed minus 27 the K cubed is K cubed, then we have minus which is the difference of part in 27. That's three cubed. For the second one here again, K cubed is the K cubed minus is the difference. And the one that's one cubed. So we recall from previous lessons that when we have a difference of cubes, which could be written as open parentheses. A cubed minus B cubed, closed parentheses, we can factor that as open parentheses. A minus B closed parentheses multiplied by open parentheses. A squared plus A B plus B squared, closed parentheses. Now, for both of these two differences of cubes, we need to identify our A S and our BS and then we can plug them in to the factor difference of cubes. So we'll end up having four sets of parentheses here. Let's first work on our K cubed minus and identify our A S and BS. So our A s here for K cubed, that's the first one being cubed. That's going to be our A term is going to be K and the B for B cubed, that's the second one being cubed, the one being subtracted. And the B term here is going to be three. And so now filling in our A S and BS for our KS and threes So I'm going to start this way over here. We've got open parentheses instead of A, that's K and then minus, instead of B, that's three closed parentheses, multiplied by open parentheses. Instead of A squared, it's going to be K squared, then plus, and then for a multiplied by B that's K multiplied by three, which is three K. And then for plus B squared, we have our plus and instead of B squared, it's three squared, three squared is nine, we can close parentheses. All right, we have factored that first difference of cubes. Now let's work on the second one identifying our A's and B's. So again, A is the first one being cubed. A is again, K B is the second one being cubed, the one being subtracted and here B is one. So now filling in our A S and BS for our KS and ones, we'll take these first two sets of parentheses that we've already factored for our K cubed minus 27. And we'll multiply those by open parentheses. Now, filling in for A minus B A is K minus B is one closed parentheses multiplied by open parentheses. A squared. That's going to be K squared plus A multiplied by B. Well, A is K B is one and M K multiplied by one is just K and then plus B squared. Well, B is one. So one squared is just one closed parentheses. And this is this whole thing fully factored. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

In Exercises 95–104, factor completely. x⁶ − 9x³ + 8

Key Concepts

Factoring Polynomials

Difference of Squares

Substitution Method

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