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.

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

Key Concepts

Sum of Cubes Formula

Identifying Perfect Cubes

Factoring Polynomials

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