In Exercises 1–10, factor out the greatest common factor. 9x^4−18...

Question

In Exercises 1–10, factor out the greatest common factor. 9x^4−18x^3+27x^2

Accepted Answer

Hello today we're going to identify the greatest common factor of a polynomial and then factoring it out. So the polynomial given to us is five X to the fourth minus 15 X cubed plus 45 X squared. So we want to go ahead and first identify what our greatest common factor is going to be. Well if we take a look at the three terms, specifically the leading coefficients, we have 5, 15 and 45. Each of these coefficients are divisible by five. So part of our greatest common factor is going to be five because we can go through and divide each term by five. Now, if we take a look at the variables we have X to the fourth X cubed and X squared. There's at least two instances of X in every term. So each term is divisible by X squared. So our greatest common factor is going to be five X squared. So with that being said, what does it mean to factor out? The greatest common factor? Well, what that means is we're gonna go through to each term and divide each term by five X squared. So dividing each turn by five X squared while five X to the fourth divided by five X squared. The fives are just going to reduce down to one and X to the fourth divided by X squared is just going to leave us with X squared negative 15 X cubed divided by five X squared negative 15 divided by five is just going to leave you with negative three and X cubed divided by X squared is just going to be X. So our second term is going to be negative three X 45 X squared divided by five X squared. Well 45 divided by five is just going to leave you with positive nine and X squared divided by X squared is just gonna cancel itself out. So our last term with five X squared factored out is just going to be nine. So the way that we write the factored form now is we put our greatest common factor in front of our neutrino meal which has been factored. So that's going to be five X squared times the quantity X squared minus three X plus nine. And this is going to be the factored version of our given polynomial. And the answer to this problem is going to be be So I hope this video helps you in understanding what it means to factor out the greatest common factor and I'll go ahead and see you all in the next video.

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