Question

Write an equation for a polynomial function (in factored form) with zeros at 3, -1, and 0. The zeros at 3 and 0 have a multiplicity of 2 and the zero at -1 has a multiplicity of 3.

Answer 1

A polynomial function in its factored form is written:

$\begin{gathered} y=(x-r_1)(x-r_2)(x-r_3)\ldots(x-r_k) \\ \text{ Where }r_1,r_2,r_3,\ldots,r_k\text{ are the zeros of the function} \end{gathered}$

On the other hand, the number of times a given factor appears in the factored form of the equation of a polynomial is called multiplicity.

So, in this case, we have

$\begin{gathered} r_1=3 \\ r_2=-1 \\ r_3=0 \\ y=(x-r_1)(x-r_2)(x-r_3) \\ y=(x-3)^2(x-(-1))^3(x-0)^2 \\ y=(x-3)^2(x+1)^3x^2 \\ \text{ Order} \\ y=\mleft(x+1\mright)^3(x-3)^2x^2 \end{gathered}$

Therefore, the equation for the polynomial function (in factored form) is

$y=(x+1)^3(x-3)^2x^2$