In Exercises 1–22, factor the greatest common factor from each po...

Question

In Exercises 1–22, factor the greatest common factor from each polynomial.

32x⁴ + 2x³ + 8x²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following polynomial were given 81. Why? To the fourth plus three? Why to the third plus 12 Y squared? And our answer choices are answer choice A three Y squared times parentheses 27 Y squared plus Y plus 12. And parentheses answer choice B Y squared times parentheses, 81 Y squared plus Y plus 12 and parentheses answer choice three wide cubed times parentheses 27 Y cubed plus Y squared plus four Y and parentheses in answer choice D three Y squared times parentheses 27 Y squared plus Y plus four. Alright. Well, we recall from previous lessons that the first step in factoring a polynomial is to look at the parts that are similar. So let's start off looking at the coefficients. We have 81 3 and 12 and we ask ourselves with the coefficients. What's the greatest common factor? What's the largest number that all three of those are divisible by? And the greatest common factor here is three. So we can divide all of them by three or in other words, factor out a three. Alright. The next part of each term that we look at to see what's similar are these wise? So they all have A Y, at least one of them. The first term has Y to the fourth, the second term has Y to the third and the third term has Y squared. And we recall from previous lessons that when we are looking at variables, we are going to factor out the variable with the lowest degree. So between 43 and two, which one is the lowest degree? That's two. So we're going to factor out A Y squared or divide them all by Y squared. Alright, we figured out what we can factor out. Now, we need to figure out what's left inside. So we're going to divide each term by what we factored out. So starting with 81 Y to the fourth divided by three, Y squared, 81 divided by three equals 27. So there's 27 left inside the parentheses there. And Y to the fourth divided by Y squared, we recall from previous lessons that when we are dividing variables, we are going to subtract their exponents. So four minus 24 minus two is two. So we're left with Y squared and there's the first term. Now let's see what's left of the second term. Three divided by three equals one. So there's plus one left of those three. And why? To the third divided by Y squared? Well, three minus two equals one. So that's why to the first or just why? So we have one times why, typically when we have one times why we wouldn't put that one there, we would just put why? And now the third term 12 Y squared divided by three Y squared. So taking 12, divided by three, that leaves us with four. So we can put plus four and Y squared divided by Y squared while two minus two equals zero. So we have Y to the zero and anything raised to the zero, power is equal to one. So we have four times one. Well, four times one is four. So we don't need to put the times one and we can just close the parentheses. Well, now we fully factor this polynomial and we want to look at our answer choices for three Y squared times parentheses, 27 Y squared plus Y plus four, end parentheses. And that matches with answer choice D while done, we'll catch you on the next one.

In Exercises 1–22, factor the greatest common factor from each polynomial. 32x⁴ + 2x³ + 8x²

Key Concepts

Greatest Common Factor (GCF)

Factoring Polynomials

Polynomial Terms

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