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. 2x⁵ + 54x²

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 are given is five M raised to the fifth power plus 40 M squared. Our answer choices are a five M squared multiplied by the factor M plus two multiplied by the factor M squared plus two M plus four B five M multiplied by the factor M plus two multiplied by the factor M squared minus two M plus four C five M squared multiplied by the factor M plus two multiplied by the factor M squared minus two M plus four or D. The given binomial is prime. First thing I want to do is rewrite my original expression. So five M raised to the fifth power plus 40 M squared. I recall my first step when factoring is I will look for a greatest common factor. So a value I can divide evenly out of all the terms involved. Looking at my coefficients of five and 40. I recognize that I can factor a five out of both. Then looking at the variables M raised the fifth power and M squared, I noticed it is possible to factor M squared out of both of those. So five M squared is my greatest common factor. I'm gonna open a set of parentheses directly next to it. And then I'm going to divide each term in my original expression by five M squared. So five M raised to the fifth power divided by five M squared. Five divided by five cancels out. Recall that when we divide like bases, we subtract our exponents. So we are left with M cubed. Next 40 M squared divided by five M squared, 40 divided by five is a positive eight M squared divided by M squared will cancel out. So at this step of factorization, we have five M squared multiplied by open parentheses. M cubed plus eight closed parentheses. We're not done factoring this just yet. In my parentheses, I have the sum of cubes eight is a perfect cube as is M cubed. This means I can use the format that I know for factoring the sum of cubes to be able to get this a little bit more broken down. So reminding you what that format looks like if we had something like X cubed plus Y cubed, we can factor this into a binomial times a trinomial. Our binomial is the cube root of both of the original terms. So the cube root be X plus Y. In this case, our trinomial is comprised of the cube root squared of the first term. So X squared, the middle term is negative and then it's the product of X Y. And the last term is a positive Y squared. So that's our general format. In our example, we have M cubed plus eight. So this means that X is the same as M for us and we need to find what the cube root of eight is. So continuing to do this off to the side my Bino meal. So we got X before because that was the cube root of X cubed. So the cube root of M cubed is M plus. We had the why before, which was the cube root of Y cubed? Here, we're looking for the cubed route of eight. So what times itself three times gives us eight. That would be two. So in our binomial, we have M plus two, our trinomial is gonna be comprised the same way. So in our example, it's X squared X is the same as M for us. So this will be M squared, the middle term is negative, then it's X times Y or X multiplied by Y. So for this, that would be two multiplied by M. So two M and the last term of the trinomial is Y squared. So we found that Y in our case is 22 squared is four. So we're gonna write plus four closed parentheses. So now that I know that M cubed plus eight equals the factor M plus two multiplied by the factor M squared minus two M plus four. I'm gonna plug it back into my original factoring. So I end up with five M squared multiplied by M plus two. Close parentheses, multiplied by the parentheses. M squared minus two M plus four. Closing those parentheses. This appears to be as factored as this can get the trinomial M squared minus two M plus four cannot be factored any further. And there's no other greatest common factors or anything else to factor. So this is our solution five M squared multiplied by the factor M plus two, multiplied by the factor M squared minus two M plus four. And that's answer choice C have a nice day.

In Exercises 1–68, factor completely, or state that the polynomial is prime. 2x⁵ + 54x²

Key Concepts

Factoring Polynomials

Greatest Common Factor (GCF)

Prime Polynomials

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