In Exercises 65–92, factor completely, or state that the polynomi...

Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^4−162

Accepted Answer

Hello today we're gonna be factoring a polynomial. So the polynomial given to us is three X to the fourth? Power minus 48. Now, in order to factor this, what we can go ahead and do is identify a greatest common factor. Well, in both of these terms they have three and 48 which are both divisible by three. So the first thing we can go ahead and do is factor out the greatest common factor of three. Doing so is going to give us three Times The Quantity X to the 4th -16. Now X to the fourth minus 16 is in the form of the difference of squares. Now the difference of square states that if you have a polynomial in the form of a squared minus B squared, this could be factored as a plus B times A minus B. Well, it's pretty easy to identify what R. B is here because 16 is a perfect square of 4. 16 is four times four. So B is going to equal to four. But what about X to the fourth? Well, X to the fourth could be rewritten as X squared times X squared because X squared times X squared is equal to X to the power of two plus two, which is equal to X to the fourth. So we're gonna go ahead and let a equal to X squared. Now, what we want to go ahead and do is put our A and B into the difference of squares. Well, we're gonna go ahead and write down our first factor which was the greatest common factor of three times a plus B or x squared plus four times a minus B, which is x squared minus four. Now, x squared plus four is not factory able but x squared minus four is factory ble because that itself is also a difference of squares. So x squared could be rewritten as x times X. So A is going to equal to X in this case and four could be rewritten as two times two. So B is going to equal to two. So writing down our first two factors is going to give us three times X squared plus four times our difference of squares of X squared minus four, which is going to be a plus B or X plus two times a minus B or x minus two. And this is going to be the fully factored version of our given polynomial. And that means that the answer to this problem is going to be be so I hope this video helps you in understanding how to factor using the difference of squares. And I'll go ahead and see you all in the next video

In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^4−162

Key Concepts

Factoring Polynomials

Difference of Squares

Greatest Common Factor (GCF)

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