how can i find the zeros

Question

Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.)

P(x) = x3 − x2 − 12x

Answer 1

Hi Lana,

I am operating under the assumption that "x3" and "x2" in your question mean x³ and x², respectively.

Firstly, note that we have a common term x which we can factor out, i.e.:

x³ – x² – 12x = x(x² – x – 12x)

So one of the factors of this polynomial is just x.

Now we must factor what remains. This is just a simple quadratic equation; we are looking for two numbers that add to -1 and multiply to -12. What we find is that 3 & -4 fit the bill.

Thus:

x³ – x² – 12x = x(x+3)(x–4)

And by the zero product property, this means we have zeroes at x = 0, x = -3, & x = 4.

Each have multiplicity 1.

If one of those binomials had a multiplicity of, say, two, it would look like for example (x+4)², where the exponent of the binomial term in the polynomial as expressed in factored form (in this case 2) is that zero's multiplicity.

In this case, though, we don't have any zeroes like that so they each have multiplicity 1.

Hope that's helpful.

Answer 2

P(x) = x(x² - x - 12)

Now factor the quadratic.