Question #45160

Feb 13, 2017

See explanation.

Explanation:

To solve this task you need to use the folloing feature of polynomial:

If $a$ is a zero of polynomial with multiplicity $n$, then the polynomial is divisible by ${\left(x - a\right)}^{n}$ and not divisible by ${\left(x - a\right)}^{n + 1}$

As stated above the polynomial would have to be divisible by ${\left(x - 3\right)}^{3}$, and ${\left(x - 0\right)}^{2}$. The only polynomial of degree $5$ fulfilling theses conditions is:

$P \left(x\right) = {\left(x - 3\right)}^{3} \cdot {x}^{2} = \left({x}^{3} + 3 {x}^{2} + 3 x + 1\right) \cdot {x}^{2}$
$= {x}^{5} + 3 {x}^{4} + 3 {x}^{3} + {x}^{2}$