In Exercises 69–78, factor each polynomial.

2y⁷(3x−1)⁵ − 7y⁶(3x−...

Question

In Exercises 69–78, factor each polynomial.

2y⁷(3x−1)⁵ − 7y⁶(3x−1)⁴

Accepted Answer

Hello, everyone. We are going to factor the following polynomial expression. The expression is seven U raised to the seventh power multiplied by the parentheses. Two V minus three, raised to the fifth power -2. U raise to the 6th power multiplied by the parentheses. Two V minus three, raised to the fourth power. Our answer choices are a U raise the sixth power multiplied by the parentheses. Two V minus three, raised to the fourth power multiplied by the parentheses. 14 U V minus 21 U plus two B seven U raise the sixth power multiplied by the parentheses. Two V minus three, raised to the fourth power multiplied by the parentheses. Two U V minus three, U minus one C. You raised the sixth power multiplied by the parentheses. Two V minus three, raised the fourth power multiplied by the parentheses. 14 U V minus 21 U minus two D. This polynomial is prime. I'm going to rewrite my original expression. seven U raised to the 7th power multiplied by the parentheses. Two V minus three, the parentheses erased the 5th power -2. U raised the 6th power multiplied by the parentheses. Two v -3 parentheses raised to the 4th power. There is a lot going on in this expression. So we are going to start by looking for greatest common factors. So the largest value that can be taken evenly out of each term. First, I'm gonna look at the coefficients, the coefficients are seven and negative two, nothing in common there. So I'm moving on to the U. So I have you to the seventh power and you to the 6th power, I can take a U to the sixth power out of both of those. So that's gonna be part of my greatest common factor. You raised to the 6th power. Next, looking at the parentheses, 2 V -3. That binomial is in both terms with different exponents. Since it's in both terms, it will be part of my greatest common factor. So I wanna see how many exponents I can take with it. And since one is raised to the fifth power and the next time it's raised to the fourth power, this means I could factor the parentheses, two V minus three, raised to the fourth power out of both terms. So now that I have the greatest common factor of six of sorry, you raised the sixth power multiplied by two V -3 in Parentheses, raised to the 4th power. I'm gonna go through and divide each term in my original binomial by my G C F. So I'm opening a set of parentheses and within those parentheses. I'm gonna start by doing seven. U raised the seventh power multiplied by the parentheses. Two V minus three, raise the fifth power divided by you, raise the 6th power multiplied by 2 V -3, raise the 4th power. So when I do that, the seven doesn't change U to the 7th divided by U to the 6th. Recall that when we divide like bases, we subtract our exponents. So that'll leave us with you to the first power or just you. And then the parentheses, two V minus three, raise the fifth power divided by the parentheses. Two V minus three, raise to the fourth power will give us parentheses 2 V -3 closed parentheses. Then the next term I'm gonna put a minus sign down two U raise the sixth power multiplied by the parentheses. Two V minus three, raise the fourth power will now be divided by U raised to the 6th Power Times 2 V -3 in parentheses, raised to the 4th power. The two has nothing to change it. So two gets written down in my parentheses. U to the 6th, divided by U to the 6th will cancel out the parentheses. Two V minus three, raised to the fourth power divided by the parentheses, two V minus three raised to the fourth power will also cancel out. So I can close my parentheses at this step. I have U raised to the sixth power multiplied by the parentheses two V minus three closed parentheses. Raise to the fourth power multiplied by parentheses. Seven U multiplied by parentheses. 2 V -3 closed parentheses -2 closed parentheses. I think I can clean this up a little bit within my second set of parentheses. I want to distribute the seven U into the parentheses. With the two V minus three, gonna rewrite the rest. So you raise the 6th power multiplied by 2 V -3 in parentheses, raised the 4th power, then open parentheses and I'm gonna distribute. So seven U multiplied by two V would give me U V seven U multiplied by - would be negative 21 U and then -2 doesn't change. So we rewrite it and close parentheses. I do not think there's anything else in here I can combine. So you raised the sixth power multiplied by open parentheses. Two V minus three, closed parentheses, raised to the fourth power multiplied by open parentheses. 14 U V minus 21 U minus two closed parentheses is our solution. And that is answer choice. C have a nice day.

In Exercises 69–78, factor each polynomial. 2y⁷(3x−1)⁵ − 7y⁶(3x−1)⁴

Key Concepts

Factoring Polynomials

Common Factor

Exponents and Power Rules

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