Question

Find a polynomial of degree 4 with -3 as a zero of multiplicity 2 and 0 and 6 as zeros of multiplicity 1

Answer 1

so we have the things we need

(x+3) will give us a zero at -3
(x-0) = x will give us a zero at 0
(x-6) will give us a zero at 6

we want the -3 one to be of multiplicity 2 so (x+3)^2
we want the other two to have multiplicity 1
x^1 and (x+6)^1

so our polynomial is \[\begin{equation}
\begin{aligned}
x(x+3)^2(x-6) &=x[(x^2+6x+9)(x-6)] \\
&=x(x^3-6x^2+6x^2-36x+9x-54) \\
&=x(x^3-29x-54) \\
&=\large \color{red}{x^4-29x^3-54x}
\end{aligned}
\end{equation} \]

Answer 2

easy peasy

Answer 3

thanks @zzr0ck3r