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. 12x³ + 3xy²

Accepted Answer

Hello, everyone. We have been asked to perform a complete factorization of the given expression or categorize it as a prime polynomial. The expression we're given is 34 M cubed plus 17 M N squared. Our answer choices are a 17 multiplied by open parentheses. Two M squared plus N squared, closed parentheses. B 17 M multiplied by open parentheses. Two M squared plus N squared, closed parentheses. C 17 M squared multiplied by open parentheses. Two M plus N squared, closed parentheses. D the given polynomial is prime. Looking back at my original expression 34 m cubed Plus 17 MN Squared. I recall that when I want to do a complete factorization, the first thing I look for is a greatest common factor. So a value that can be divided evenly out of all terms in the expression starting by looking at my coefficients 37 and 7 or sorry, 34 and 17. I wanna see can both of those values be divided by the same number and they can both be divided by 17. So 17 is part of my greatest common factor. Next, looking at the variables, I noticed both terms have an M one M cubed and one is just M, which means I could factor out M to the first power. 17 M is my greatest common factor that I'm gonna multiply it by open parentheses. And what goes in the parentheses is the values that are left when I divide each original term by my greatest common factor. So 34 M cubed divided by M, 34 divided by 17 is 2 M cube divided by M to the first power. We keep our like base, we subtract our exponents. So we get M squared. Our next term 17 M N squared divided by 17 M 17, divided by 17 is one which we don't need to write down M divided by M will cancel. And we're left with N squared closed parentheses. So at this point in my factorization, I have 17 M multiplied by the parentheses. Two M squared plus N squared, I want to see is there anything else that can be factored? I know in my parentheses, I have two squares but two is not a perfect square. And that's addition, we can only factor the difference of two squares. So this looks like it's as factors that factored as it can be. And that's answer choice. B have a nice day.

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In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³ + 3xy²

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