Factorisation of polynomials
Which monomial factorization is correct?
- "Theodore and Claire were each asked to factor "the term 24 x to the fifth as the product of "two monomials. "Their responses are shown below." So Theodore factored 24 x to the fifth as being equal to eight x third, times three x squared, and Claire factored 24 x to the fifth as being equal to four x times six x to the fourth. And then they ask us, "Which of the students "factored 24 x to the fifth correctly?" So I encourage you, pause the video and see if you can figure this out. Which of them factored it correctly? All right, now let's first look at Theodore. So he factored it into these two monomials, eight x to the third and three x squared. Well, let's just see, if we were to multiply these two things, do we get 24 x to the fifth? So if you multiply eight times three, you do indeed get 24. And then all you have to do is multiply the x terms, or the powers of x. You have x to the third times x squared, that indeed does equal x to the fifth. So Theodore did factor it correctly, this is one factorization, I guess you would say, of 24 x to the fifth. Now let's look at Claire. So Claire, if we were taking just the coefficient, four times six is indeed equal to 24. And then if we were to look at the powers of x we have x to the first power here, times x to the fourth power, which is going to be x to the fifth power. So Claire also factored it correctly. And this just goes to show you that there's more than one possible factorization of a monomial like 24 x to the fifth. I could come up with another one. I could write something like 24 x to the fifth, I could say that that is 12 x, to the third, times, what would have to be left? 12 times two is 24, so two x squared. That's another possible factorization. So clearly there's more than one way to factor this monomial into two other monomials.