# Thread: Need Help. Polynomials and multiplicity

1. ## Need Help. Polynomials and multiplicity

Can someone help me with the problem. I need to understand it. I'm having a hard time creating the polynomial. The roots of multiplicity and the x=2 and x=0 are throwing me off. Where are they getting these from???

Here is the question:

The polynomial of degree 5, ${P}{\left({x}\right)}$ has leading coefficient 1, has roots of multiplicity 2 at ${x}={2}$ and ${x}={0}$, and a root of multiplicity 1 at ${x}=-{4}$
Find a possible formula for ${P}{\left({x}\right)}$.

Thank you,

Turquoise Bird

2. ## Re: Need Help. Polynomials and multiplicity

Originally Posted by TurquoiseBird
Can someone help me with the problem. I need to understand it. I'm having a hard time creating the polynomial. The roots of multiplicity and the x=2 and x=0 are throwing me off. Where are they getting these from???

Here is the question:

The polynomial of degree 5, ${P}{\left({x}\right)}$ has leading coefficient 1, has roots of multiplicity 2 at ${x}={2}$ and ${x}={0}$, and a root of multiplicity 1 at ${x}=-{4}$
Find a possible formula for ${P}{\left({x}\right)}$.

Thank you,

Turquoise Bird

So it has a root at x = 2, which means x - 2 = 0. Therefore, x - 2 is a factor. It's also repeated, so (x - 2)^2 is a factor.
Similarly, it has a root at x = 0, which means x is a factor. It's repeated, so x^2 is a factor.
Similarly, it has a root at x = -4, which means x + 4 = 0. Therefore, x + 4 is a factor.

So P(x) = x^2(x - 2)^2(x + 4) is a possible formula. So is any constant multiple.

3. ## Re: Need Help. Polynomials and multiplicity

Multiplicity of a root is basically the number of times a particular number is a zero for some polynomial. In your case, for instance, $(x - 2)^2 = 0$ means that 2 is a root and because it is squared we say it has multiplicity of 2. If the term were cubed it would have multiplicity of 3 and if it was not raised to a power it would have multiplicity of 1. To answer the question of where they get it from in this problem they are just making it up so that you can write down the equation.