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. 11x⁵ − 11xy²

Accepted Answer

Hello, everyone. We've been asked to perform a complete factorization of the given expression or categorize it as the prime polynomial. The expression we are given is 18 M raised to the fifth power minus 18 mn squared. Our answer choices are a 18 multiplied by open parentheses. M squared minus N closed parentheses multiplied by open parentheses. M squared plus N closed parentheses. B 18 M multiplied by open parentheses. M squared minus N closed parentheses multiplied by open parentheses. M squared plus N closed parentheses. C M multiplied by open parentheses. 18 M squared minus N closed parentheses multiplied by open parentheses. 18 M squared plus N closed parentheses. D the given binomial is prime. Going back to our binomial. We have 18 M raised the fifth power minus 18 mn squared. Recall that any time you're asked to do a complete factorization. The first thing you should look for is a greatest common factor or a value that you can divide evenly out of all terms. In this case, I will look at my coefficients first noticing that both coefficients are 18. I know that 18 will be part of my greatest common factor or GCF. Next looking at the variables I have M to the fifth in my first term and M in my second term, which means I can factor an M out of both terms. So 18 M is my greatest common factor. And I'm gonna multiply that by a set of parentheses and in those parentheses, I'm gonna make terms from dividing each term in the original expression by 18 M. So 18 M raised to the fifth power divided by 18 M, 18, divided by 18 will cancel out M raised to the fifth power divided by M to the first power. Give us M raise to the fourth power. Recall that when you have like bases and you divide, you subtract your exponents next negative 18 M and squared divided by 18 M will result in a negative and then the eighteens will cancel out. The MS will cancel out and will be left with N squared close parentheses. So at this step, we have 18 M multiplied by open parentheses. M raised to the fourth power minus N squared closed parentheses. This can be factored further because what's in my parentheses is the difference of two squares M to the fourth power is the same as saying M squared squared. So I'm gonna factor this as the difference of two squares. So I'm opening two sets of parentheses next to each other that are currently empty. I recall to find the values that go in these binomials for the first term in each binomial, I take the square root of the first term of difference of two squares. So the square root of M raised to the fourth power is M squared. So that is the first term in each of my new binomials. The second term in each of these binomials comes from the square root of the second term in the difference of two squares. So the square root of N squared, which is N will be the second term in each binomial. And then I recall that they must have opposite signs. So one set of parentheses has a minus sign. The next has a plus sign. This reads as 18 M multiplied by open parentheses. M squared minus N closed parentheses multiplied by open parentheses. M squared plus N closed parentheses. Since I do not have another difference of squares and I do not have any other greatest common factors. This is my final factorization and it's answer choice B have a nice day.

In Exercises 1–68, factor completely, or state that the polynomial is prime. 11x⁵ − 11xy²

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Prime Polynomials

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