Factor each polynomial completely. See Example 6.t⁴ - 1

Question

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 K raised to the fourth power minus 16. Our answer choices are answer choice. A, the quantity of K squared plus four multiplied by the quantity of K plus two multiplied by the quantity of K minus two. Answer choice B, the quantity of K squared plus four multiplied by the quantity of K plus two squared. Answer choice C, the quantity of K squared plus four multiplied by the quantity of K minus two squared. And answer choice D, the quantity of K squared plus two multiplied by the quantity of K plus two multiplied by the quantity of K minus two. All right. So looking at our expression, when we start to think about how to factor this breaking it down into its factors, we recognize that we have a difference of squares. I'm recalling from previous lessons that a difference of squares is expressed when we have a squared. So something squared minus B squared. So minus something else squared and we factor this to B so we set this equal to the quantity of A plus B multiplied by the quantity of A minus B. So a lot of this is just figuring out what's our A, what's our B and then plugging it in. So for this first one, we see that the K raised to the fourth power is our A squared. So what's being squared for K to the fourth power that's going to be K squared. So A here equals K squared and now our B squared, that's what's being subtracted. What's being squared to equal 16? That's going to be four. So B here equals four. Now we can plug this in, in our factored form. We have open parentheses. A is K squared plus B is four multiplied by the quantity of and our A again is K squared, but now this time minus our B is four. And so this looks so wonderful and nice. Our K squared plus four multiplied by K squared minus four. But we're not done because we have another difference of squares. This K squared minus four is a difference of squares as well. So we see that our K squared is our A squared. What's being squared to equal K squared? That's going to be K. So A here, this time is K and B squared, that's what's being subtracted B squared. Here is four. What's being squared to get four? That's two. So B equals two. Now we can plug in for A and B in our factored form. We have open parentheses. A is K. So K plus B is two closed parentheses multiplied by. And now open parentheses, K again is A, this time minus B again is two close parentheses. So now that is the factored form of this second difference of squares. So if we were to bring all of these factors together, we have open parentheses. K squared plus four closed parentheses multiplied by the quantity of K plus two, multiplied by the quantity of K minus two. And that's as factored as these get, we don't have a sum of squares factorization available and that's it. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Factor each polynomial completely. See Example 6.t⁴ - 1

Key Concepts

Factoring Polynomials

Difference of Squares

Further Factoring

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