Factor each polynomial. See Examples 5 and 6. 125-(4a-b)^3

Question

Accepted Answer

Hello, today we're going to be factoring the given expression. So what we are given is 343 minus and minus three A cubed. Now, before we start factoring this, there's one change that we can make to this expression. And that's the fact that 343 is equal to seven cubed. So if we replace 343 with seven cubed, what we get is seven cubed minus the quantity M -3 A Cubed. Now, what we have here is a difference of cube terms and there's a special factor for that. The special factor states that if you have X cubed minus Y cubed, this can factor to the quantity X minus Y times the quantity Y X squared plus X times Y plus Y squared. So here we can let x equal to seven and we can let y equal to the quantity M -3 a. And if we substitute X and Y into the special factor for X cubed minus Y cubed, what we get is X minus Y which is seven minus the quantity M minus three A times X squared, which is seven squared. Plus X times Y which is seven times M minus three A plus Y squared, which is going to be M minus three A squared. Now, there's a bit of simplification that we can go ahead and do to this expression here. First, we can go ahead and distribute negative one into M -3 a Seven squared is going to evaluate to 49 and we can go ahead and distribute seven into M -3 a. Doing these simplifications is going to give us seven minus M plus three A times plus seven times M minus three A. So seven times M is going to give us seven M and seven times negative three A is going to give us negative 21 A and that's gonna be plus M minus three A squared. So next, we're going to need to simplify M minus three A squared. And we can do that by foiling the quantity out C M minus three A squared is the same thing as saying M minus three A times M minus three A. So in order to foil this, we're going to distribute em into em and negative three A and we're going to multiply negative three A two A M and negative three A doing this is going to give us M times M which is M squared, M times negative three A which is negative three M A negative three times M which is going to give us another negative three M A and negative 38 times negative three A, which is going to give us positive nine A squared. So we can go ahead and combine like terms here by combining the two negative three M A s together. And what we get is M squared minus six M A plus nine A squared. So if we did substitute this value for M minus three A squared, what we get is the expression seven minus M plus three A times the quantity four A plus seven M minus 21 A plus the quantity M squared minus six M A plus nine A squared. And this is going to be the completely factored form of our given expression. And with that being said, the answer to this problem is going to be be. 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. 125-(4a-b)^3

Key Concepts

Factoring Polynomials

Difference of Cubes

Binomial Expansion

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