In Exercises 95–104, factor completely.8x⁴ − x/8

Question

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 27 D raise to the fourth power minus D divided by 27. And we could rewrite the second part as minus one, 27th multiplied by D and then bring down the 27 D raise to the fourth power. Our answer choices are answer choice ad multiplied by open parentheses. 3d minus one third, closed parentheses multiplied by open parentheses. Nine D squared minus 3d plus 1/9 closed parentheses. Answer choice B is D multiplied by open parentheses, 3d minus one third, close parentheses multiplied by open parentheses, nine D squared plus 3d minus 1/9 closed parentheses. Answer choice C is D multiplied by open parentheses 3d minus one third, closed parentheses multiplied by open parentheses. Nine D squared plus D plus 1/9 closed parentheses. In answer choice D is that this polynomial is prime. Well, the first step when multiplying or when factoring, excuse me is that we're going to look and see if we have any common factors between our terms and looking at these two terms both of them have ad variable. So we can factor ad out in front and then we divide both terms by D and see what we have left. So we have D multiplied by 27 d rays to the fourth power divided by D will leave us with 27 d cubed. And then we have minus and one 27th D divided by D will leave us with just one 27th and we can close those parentheses. All right. So now we've factored out that D and we notice that inside of our parentheses, we have a difference of cubes. 27 is three, cued. D is D cubed is D cubed and we have the minus, which is the difference of part and one 27th is one third cubed. So we recall from previous lessons that when we have a difference of cubes, we could write that as open parentheses. A cubed minus B cubed closed parentheses. And that will be factored as open parentheses. A minus B closed parentheses multiplied by open parentheses. A squared plus A B plus B squared closed parentheses. Now, we just need to identify our A and B terms, plug those into the factored difference of cubes and then we will have our expression fully factored. So first let's identify our a term. Our A term is the first one being cubed. Well, here that's going to be 3d. Both of those are cubed for the first one and now our B term, that's the one being subtracted, the second one being cubed and here B is equal to one third. And now we can plug in 0D for a and 1/3 for B before we do that, I want to bring down this D that we factored out at the very beginning. I don't want to forget that. So we have ad multiplied by open parentheses. Now, this will be our A minus B A is going to be 0D and then we have minus and our B is one third, close those parentheses multiplied by open parentheses. Now we have A squared. Well, A is 3d. So 3d squared, the three squared is nine and D squared is D squared. Then we have plus our, A multiplied by our B, our A is 3d multiplied by our B which is one third and we take three multiplied by one third and that's just one. So this is just leaving us with AD so plus D and now we have our B squared, our B is one third and one third squared is 1/9. So plus 1/9 close those parentheses. And now this has been fully factored. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

In Exercises 95–104, factor completely.8x⁴ − x/8

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Polynomial Degree and Terms

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