Factor each polynomial. See Examples 5 and 6. 125x^3-27

Question

Accepted Answer

Hello, today we're going to be factoring the given expression. So what we are given is 343 Q cubed minus eight. Now, there's two changes that we can make to the current expression. 343 is equal to seven cubed. So we could replace 343 with seven cubed and eight is equal to two cubed. So we could replace eight with two cubed making the substitution is going to give us seven cubed times Q cubed minus two cubed. And if we use the rule for exponential distribution, which states that if you have eight times B raised to the power of three, you can just go ahead and distribute this exponents to each term in the product. And that's going to give you a cubed times B cubed. If we use this rule backwards on seven cubed times Q cubed, we can go ahead and combine the quantity together to give us seven Q cubed minus two cubed. Now, what we have here is a difference of cube terms and we have a special factor for the difference of cube terms. The special factor states that if you have X cubed minus Y cubed that is going to equal to X minus Y times the quantity X squared plus X times Y plus Y squared. Here, we're going to let X equal to seven Q and we're going to let Y equal to two. So if we plug in these values of X and Y into the special factor for a difference of cubes, what we get is X minus Y which is seven Q minus two times the quantity X squared, which is seven Q squared plus X times Y, which is seven Q times two plus Y squared, which is going to be two squared. So we need to go ahead and do a little bit of simplification. We know that two squared is going to evaluate to give us four, seven Q times two is going to give us 14 Q and seven Q squared. While in order to simplify this, we're going to need to use our exponential distribution and we can distribute the exponent of two to both seven and Q. So what we get is seven squared times Q squared and seven squared is going to give us 49. So we have 49 Q squared. So once we plug in these values, what we get is seven Q minus two times 49 Q squared Plus Q Plus four. And this is going to be the completely factored version 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 factor this expression. And I'll go ahead and see you all in the next video.

Factor each polynomial. See Examples 5 and 6. 125x^3-27

Key Concepts

Factoring Polynomials

Difference of Cubes

Polynomial Degree

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