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. 12x³y − 12xy³

Accepted Answer

Hello, everyone. We are asked to perform a complete factorization of the given expression or categorize it as the prime polynomial. The expression we're given is 17 M cubed, N minus 17 M N cubed. Our answer choices are a 17 M N multiplied by the parentheses. M minus N squared B 17 M N multiplied by the parentheses. M plus N multiplied by the parentheses. M minus N C 17 M N multiplied by the parentheses. M plus N closed parentheses squared D, the given polynomial is prime rewriting the original expression. Again, we have 17 M cubed, N minus 17 M N cubed. Recall that the first thing I want to do when I try to factor any expression is see if there is a greatest common factor that I can take out of both terms. In this case, looking at my coefficients, I notice they are both 17s which means 17 is gonna be part of my greatest common factor or G C F. Next, I notice both terms have an M so I'm gonna take one M out. So I have M to the first power just M, I also noticed both terms have an N So I'm gonna factor out an N. So my greatest common factor is 17 M N, that's gonna be on the outside of my parentheses and inside my parentheses, I'm gonna get the values received when I divide both original terms by 17 MN. So 17 M cubed N divided by 17 M N would leave me with M squared minus 17 M N cubed divided by 17 M N will leave me with N squared closed parentheses. So right now I have 17 M N multiplied by the parentheses. M squared minus N squared. Before I say that we are done factoring this. I need to see if there's anything else that can be factored. I notice my parentheses. M squared minus N squared are the difference of two squares. So this means I can factor this further. So I'm rewriting 17 MN and then I'm gonna open two sets of empty parentheses because I recall that the difference of two squares factors from one binomial into two. So when I'm factoring the difference of two squares, the first term in my new binomials comes from the square root of the first term in my two squares. So the square root of M squared is M, that's the first term in each binomial. The second term comes from the square root of the second term of the original binomial. So the square root of N squared, which is N and I recall that I need to have A plus sign in one binomial and a minus sign in the other. So what this looks like is 17 M N multiplied by the parentheses, M plus N multiplied by the parentheses. M minus N. There's nothing further, I could factor here. So that is my complete factorization and it's answer choice B have a nice day.

In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³y − 12xy³

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Polynomial Degree and Terms

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