Factor each polynomial completely. See Example 6.8t³ + 125

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 27 D cubed plus 343. Our answer choices are answer choice. A, the quantity of nine D plus seven multiplied by the quantity of 3d squared minus 21 D plus 49. Answer choice B the quantity of seven D plus three multiplied by the quantity of nine D squared plus 21 D plus 49. Answer choice C, the quantity of seven D plus nine multiplied by the quantity of 3d squared plus 21 D plus 49. And answer choice D, the quantity of 3d plus seven multiplied by the quantity of nine D squared minus D plus 49. All right, we take a look at our expression and we notice that this is a sum of cubes. Everything here is a cube. And we recall from previous lessons that when we have a sum of cubes, which would be written as a cubed plus B cubed, that can be factored as the quantity of A plus B multiplied by the quantity of A squared minus A B plus B squared. So now it's just a matter of identifying our A and B terms and plugging them in where they need to go. So a cubed, that's going to be the first thing being cubed, that's 27 D cubed. Now we can look at those separately. The 27 and the D cubed 27 is a cube of what, 27 is three cubed. So A equals the first part is three. And now for the D cubed D cubed is D cubed. So the cubed root of that is D A equals 3d. Now, how about B cubed B cubed is what's being added? B cubed is 343. Well, what's being cubed in order to get 343? That is seven. So our B is seven. Now, we just need to plug those in for A and B. So we'll have an open parentheses and then our A is 3d, bring down plus B is seven, close parentheses multiplied by open parentheses. A squared. We can put in parentheses for now A is 3d and we'll need to square that minus. Now, a multiplied by B. We can put that in parentheses. A is 3d, multiplied by B is seven, then bring down plus and for B squared, that's going to be seven squared. Now, we just need to simplify that second set of parentheses. So we can bring down our first set of parentheses 3d plus seven that's multiplied by the quantity of, for 3d squared, three squared is nine and D squared is D squared minus 3d, multiplied by seven. That's going to be 21 D so minus D and then plus seven squared is 49. And that is our expression fully factored. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Factor each polynomial completely. See Example 6.8t³ + 125

Key Concepts

Sum of Cubes Formula

Identifying Perfect Cubes

Polynomial Factoring Techniques

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