In Exercises 1–68, factor completely, or state that the polynomia...

Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁴ − xy³ + x³y − y⁴

Accepted Answer

Perform complete factorization of the given expression or categorize it as the prime polynomial where our expression is four P to the fourth minus four P Q to the third plus five P to the third times Q minus five Q to the fourth. To factor this, we will make use of grouping the grouping just states that we can group our first two terms together and our last two terms together to factor out anything that's uncommon between the two. So if we group these first two times together in the last two, let's try to factor out things that are uncommon between these first two terms. If we look here, we can see four P is in both of the terms. Let's take four P out to get four P multiplied by P to the third minus cuts dessert. Same thing for the last two terms. We can factor out five Q that leaves us with P to the third minus Q to the third in the parentheses. We then notice here P to the third minus P to the third appears twice. We can combine those into one term Multiplied by the remainders, four p plus five Q now that we're here, we can make use of the difference of cubes identity, which states if we have something X to the third minus Y to the third. This is the same thing as saying X minus Y multiple by X squared plus X Y plus. Why we, so we have P to the third minus Q to the third. That can be rewritten. We'll put four P plus five Q in the front P minus Q multiply it again by a piece of a second plus P Q plus Q squared. We can't simplify this any farther. So the answer to our problem corresponds with answers. C OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁴ − xy³ + x³y − y⁴

Key Concepts

Factoring Polynomials

Grouping Method

Prime Polynomials

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