In Exercises 75–94, factor using the formula for the sum or diffe...

Question

In Exercises 75–94, factor using the formula for the sum or difference of two cubes. 8y³ + 1

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factories. The expression completely. The expression were given is 64 P cubed plus one. And our answer choices are answer choice. A open parentheses, four P minus one closed parentheses multiplied by open parentheses. 16 P squared minus four P plus one closed parentheses. Answer choice B is open parentheses. Four P plus one closed parentheses multiplied by open parentheses. 16 P squared minus four P plus one closed parentheses. Answer choice C open parentheses, four P minus one closed parentheses multiplied by open parentheses. 16 P squared plus four P plus one, close parentheses and answer choice D open parentheses, four P plus one, close parentheses multiplied by open parentheses. 16 P squared minus four P minus one, closed parentheses. Now looking at this expression, the first thing we always do when factoring is we look at all of our terms and see if there's anything we can factor out in front and 64p cube and one do not share any common factors other than one. And we can't really factor out of one. So that step is done. And now we would ask ourselves if we have any situations here, like a difference of squares or sum of cubes or a difference of cubes. And yeah, that is what we're dealing with here. We've got 64 which is four cubed and then we've got P cubed and one, well, one can be whatever exponents we want it to be here. We can have one cubed as well. So this is a sum of perfect cubes. And we recall from previous lessons that when we have a sum of cubes that would be expressed as open parentheses. A cubed plus B cubed, close parentheses, we can factor that and that will be open parentheses. A plus B closed parentheses multiplied by open parentheses. A squared minus A B plus B squared, close parentheses. Now, all we have to do is identify our A's and B's from our original expression and plug those in to our factored some of cubes. So what's our, a term? That's the first one that's being cubed. And here, the first thing term that's being cubed or the four and the P or four P and the B that's the second one being cubed. That's a one. So B equals one. Now, we can fill these in to our factored some of cubes. So we have open parentheses instead of A, we have four P and then bring down plus and then instead of B, we have one close parentheses multiplied by open parentheses. And then A squared, that's going to be a four P squared and then we have a minus and then we have a times B, well, that's going to be four P times one, which is just four P and then plus B squared, B squared is one squared, which is just one and closed parentheses. And now we only have a little bit more work to do. We just have this four P that needs to be squared. So I'm going to bring down the first set of parentheses. We have open parentheses, four P plus one, close parentheses and four P squared. Well, the four squared is 16. So excuse me, we have this multiplied by open parentheses. Four square 2, 16 P squared is P squared. Bring down minus four P plus one close parentheses and there is our factored some of cubes. So 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 75–94, factor using the formula for the sum or difference of two cubes. 8y³ + 1

Key Concepts

Sum of Cubes Formula

Identifying a and b

Factoring Process

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