Question

How do you find a polynomial function that has zeros $x=-5,1,2$ and degree $\mathrm{n}=4$.

Answer 1

Use the given roots to construct the polynomial:

We are given 3 roots of the polynomial $x=-5,1,2$

Therefore the factors can be given as $(x+5)(x-1)(x-2)$

On expanding it can be given as $x^3+2x^2-13x+10$

But this is a cubic polynomial.

In the question a polynomial of degree 4 is asked.

Therefore one of the root has multiplicity 2.

For our convinience we assume that 1 is repeated two times

Hence

$$\begin{gathered} (x-1)(x^3+2x^2-13x+10) \\ =x^4+2x^3-13x^2+10x-x^3-2x^2+13x-10 \\ =x^4+x^3-15x^2+23x-10 \end{gathered}$$

Hence the polynomial can be given as ${\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{15}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{23}\mathit{x}\mathbf{-}\mathbf{10}$.