In Exercises 65–74, factor by grouping to obtain the difference o...

Question

In Exercises 65–74, factor by grouping to obtain the difference of two squares. x⁴ − x² − 2x − 1

Accepted Answer

Hello, today we're gonna be factoring the given polynomial. So what we are given is V to the fourth minus V squared minus six V minus nine. And we are told that we're going to be factoring this by using the difference of two squares. So what we're going to do is we're going to group up the last three terms together and we want the leading coefficient to be positive. So we're going to factor out a negative from this group doing so is going to give us V to the fourth minus the quantity V squared plus six V plus nine. And this is going to factor to give us the quantity V plus three squared. So we have V to the fourth minus V plus three squared. And one thing to know about V to the fourth V to the fourth can be rewritten as V squared times itself two times. Or in other words, it can be rewritten as V squared race to the power of two. So if we rewrite this first term as V squared race to the power of two, we get V squared raised to the power of two minus the quantity V plus three squared. So what we have is a difference of two quantities raised to the power of two. And we can go ahead and factor this by using the difference of squares, the difference of square states that if you have a squared minus B squared, then this is going to factor to give us a minus B multiplied by A plus B. Here, we're going to allow A to equal to V squared and we're going to allow B to equal to V plus three. So if we plug this into the difference of squares equation, what we end up with is the quantity A minus B which is going to be V squared minus the quantity V plus three multiplied by A plus B which is going to be V squared plus V plus three. In the first group, we're going to distribute the negative one into V plus three. And that is going to give us V squared minus V minus three multiplied by V squared plus V plus three. And this is going to be the completely factored form of the given polynomial. And with that being said, the answer to this problem is going to be C So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 65–74, factor by grouping to obtain the difference of two squares. x⁴ − x² − 2x − 1

Key Concepts

Factoring by Grouping

Difference of Two Squares

Polynomial Expressions

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