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.

15x²ⁿ − 25xⁿ

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following polynomial. We are given 26 A raised to the two N minus 52 A to the N. And our answer choices are answer choice A A to the N multiplied by open parentheses. A squared minus two closed parentheses. Answer choice B 26 A raise to the N multiplied by open parentheses. A minus two closed parentheses. Answer choice C 26 A raise to the N multiplied by open parentheses. A raise to the N minus two. A close parentheses and answer choice D 26 A race to the N multiplied by open parentheses. A raise to the N minus two closed parentheses. Well, the first step when we're factoring is that we are going to look at all of our terms and see if they have any common factors that we can factor out. So first, we'll look at the parts of each term that are alike. We'll look at the coefficients 1st 26 and 52. And we ask ourselves, what is the greatest common factor between 26 and 52 and the greatest common factor is 26. So we can divide both of them by 26 or in other words, factor out a 26. Next, we'll take a look at the variables for each of our terms. We have a raised to the two N and A raised to the N for variables when they have a like base, which is this A, we are going to look at their exponents and we want the exponent with the lowest degree, we could think of a raised to the N as a, raised to the one N. So between two N and one N, which one is lower, that would be one N. So we're going to divide both of these terms by a raised to the one N or in other words, just factoring out A raised to the N. All right. Now that we've figured out what we're factoring out in front, we want to figure out what's left behind once we've done the division. So we're going to take the 26 a race to the n multiplied by open parentheses and inside of our parentheses will be what's left from each of the two original terms. Once we divide it by 26, a race to the end and we'll divide the parts that are alike that we've already set up here. So for 26 A raise to the two N divided by A to the N, we take 26 divided by 26 which is one and that goes here inside the parentheses. And then we take a rays to the two N divided by a rays to the one N. We recall from previous lessons that when we are dividing like bases with exponents, we are going to subtract their exponents from each other. So we're taking two N minus one N which will leave us with one N. So our new exponent is one N, we have a rays to the one N which is a raised to the end. So that's being multiplied by this one. We typically don't put a one when it's a coefficient. So we'll just put a raised to the N here inside of the parentheses for our first term in the parentheses. All right, we've divided the first term by what we factored out. Now let's divide the second term. We take this minus, we can bring that down. And then we have 52. A race to the N divided by 26. A raise to the N 52 divided by 26 is two. So we have a two here for the second term in our parentheses. And then a raised to the one N divided by a raise to the one N. We're taking one N minus one N. So our new exponent is zero. We have a raised to the zero and we recall that anything raised to the zero power is just equal to one. So we have two times one, which is just two. So we don't have to put the times one there and we can close our parentheses. And now we have fact are polynomial and we look at our answer choices for 26 A to the N multiplied by open parentheses. A to the N minus two closed parentheses. And that matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 1–22, factor the greatest common factor from each polynomial. 15x²ⁿ − 25xⁿ

Key Concepts

Greatest Common Factor (GCF)

Factoring Polynomials

Polynomial Terms

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