In Exercises 57–64, factor using the formula for the sum or diffe...

Question

In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125

Accepted Answer

Hello, today we're going to be factoring the expression using the sum or difference of two cubed terms. So let's quickly recall what the factory ization of a sum or difference of cubes is if we are factoring a sum of cube terms, which can be rewritten as A Q plus B cubed. This is going to factor to A plus B times the quantity A squared minus A times B plus B squared. And if we are factoring a difference of cubes, which can be rewritten as a cubed minus B cubed, then this can be factored to A minus B times the quantity A squared plus A B plus B squared. So depending on whether you're factoring a sum or difference. Notice that the pattern here is going to be that for some, we put the plus sign first in the first quantity, then it's a minus plus. And for a difference of cubes, we put the negative sign first in the first quantity, then followed by two positive signs. So that's a trick that you can use to distinguish the difference between a sum and difference of cubes. Now let's take a look at the given expression So what we are given is 27 x cubed plus 64, 27 x cubed can be rewritten as three x times itself three times. And that itself can be rewritten as three X cubed. Next 64 can be rewritten as four times itself three times. And that can be rewritten as four cubed. So rewriting the given expression, we can write it as three X cubed plus four cubed. And now notice that we're going to factor quantity with two cubed terms. Here, we're going to allow a two equal to three X and we're going to let B equal to four. And now are we going to be factoring a sum or difference of cubes? Well, in the given expression, we are given a sum of cube terms. So we're going to need a factor using the sum of cube terms. And what that states is that we can go ahead and rewrite the expression as A plus B, which is going to be three X plus four times the quantity A squared, which is three X squared minus A times B which is three X times four plus B squared, which is going to be four squared. And now there's a little bit of simplification that we can do in the first term of the second quantity, we can distribute the exponent of two to both X and three. And this can be rewritten as nine X squared. Next three X times four is going to give us 12 X as we just need to multiply the coefficients together. So three X times four is going to give us 12 X and finally, four squared is the same thing as saying four times four, which is going to give us 16. So that means that the factory ization of three X cubed plus four cubed is going to be three X plus four times nine X squared minus 12 X plus 16. And this is going to be the factored form of the original expression. And with that being said, the answer to this problem is going to be D so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125

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