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.

x³ − 16x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to perform complete factorization of the given expression or categorize it as the prime polynomial. The expression we're given is M cubed minus 25 M. Our answer choices are answer choice am multiplied by open parentheses. M minus five, close parentheses. Squared. Answer choice B M multiplied by open parentheses. M plus five closed parentheses, squared. Answer choice C M multiplied by open parentheses. M minus five closed parentheses multiplied by open parentheses. M plus five closed parentheses and answer choice D, the given polynomial is prime. The first step in factoring is that we look at all of our terms and we see if there's any common factors that we can factor out in front. And both of these terms have an M variable so we can factor out an M and then that will leave us with, we have M multiplied by open parentheses. We take the first term M cubed and divided by M that will leave us with M squared as the first term inside of our parentheses. And then we have our minus, we can bring that down and then we have 25 M divided by M. And that will leave us with just 25. And then we close parentheses and now we've begun to factor this, but we look at what's inside the parentheses and we recognize that this is a difference of squares. They are both squares here. M squared is M squared. And then that's minus, that's the difference of and then 25 that's five squared. We recall from previous lessons when we have a difference of squares that could be written as open parentheses. A squared minus B squared closed parentheses. We can factor that as open parentheses. A plus B closed parentheses multiplied by open parentheses. A minus B closed parentheses. Now we just have to identify our A and B terms fill those in to the factor difference of squares and then we'll have our polynomial factored. So we have for our, a term that's the first one being squared. That's going to be M. So A equals M and then our B term, that's the second one being squared, the one being subtracted. And in this case, our B is equal to five. So now I am going to, this is a very important step. Bring down this M that is in front of our parentheses containing M squared minus 25. So we'll have an M and then that will be multiplied by open parentheses. And now we are going to write our difference of squares factored, filling in our A with M and our B with five. So we'll have M multiplied by open parentheses. Instead of a, we'll have M then bring over A plus. And instead of B, we'll have five closed parentheses multiplied by open parentheses. Instead of A, we have M now minus and instead of B, we have five closed parentheses, and this is our polynomial fully factored. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 16x

Key Concepts

Factoring Polynomials

Difference of Squares

Prime Polynomials

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