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.

x⁴ - 16

Accepted Answer

Welcome back. I am so glad you're here. We are asked to completely factories the following polynomial expression. If possible. Otherwise declare if the expression is a prime polynomial, our expression were given is T to the fourth power minus 625. Our answer choices are answer choice. A open parentheses, T minus five, close parentheses multiplied by open parentheses. T plus five. Close parentheses, open multiplied by open parentheses. T plus 25 close parentheses squared. Answer choice B is open parentheses, T minus 25 close parentheses squared multiplied by open parentheses. T plus 25. Close parentheses squared. Answer choice C is open parentheses, T minus five, close parentheses multiplied by open parentheses, T plus five, close parentheses multiplied by open parentheses. T squared plus 25 close parentheses and answer choice D is that this polynomial is prime. Alright. The first thing we do when factoring polynomial as we take a look at all of our terms and ask if there's anything, any greatest common factors that we can factor out and T to the 4th and 625 those do not share any common factors. So nothing we can factor out in front. The next thing we look at is we notice that we have a difference of squares here. Both t to the 4th and 625 are products of squares. So T to the fourth, that's the same as T squared squared and 625 that's the same as 25 squared. They're being subtracted. So that's a difference of squares. We recall from previous lessons that our difference of squares when we have open parentheses, A squared minus B squared, closed parentheses can be factored as open parentheses. A plus B closed parentheses multiplied by open parentheses. A minus B closed parentheses. Now, if we identify our A and R B terms, the A term, that's the first one getting squared here, that's going to be T squared. And then our B term, that's one being subtracted, the second one being squared and that's going to be 25. So now we can set up two sets of parentheses for R T to the fourth minus 625 and will factor this difference of squares. So we have open parentheses. Now, instead of A, we've got T squared and then bring over the plus and instead of B, we have 25 close parentheses multiplied by open parentheses again with A, which is T squared this time minus. And then, instead of B, it's 25 close parentheses, and we've got this pa partially factored, but we noticed we actually have another difference of squares here with this T squared minus 25 the T square that's T squared and the 25 is five squared. So are a term now is going to be T and our B term now is going to be five. So let's factor again, will bring down the first set of parentheses, open parentheses, T squared plus 25 close parentheses multiplied by open parentheses. And now this is going to be, instead of a, it's t bring over plus and instead of be it's five close parentheses multiply by open parentheses instead of a, it's T now minus and instead of be it's five close parentheses and that's as factored as this one is going to get, we look at our answer choices and this matches with answer choice. C 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. x⁴ - 16