# How do you write a polynomial function of least degree that has real coefficients, the following given zeros -5,2,-2 and a leading coefficient of 1?

Jun 28, 2016

The required polynomial is $P \left(x\right) = {x}^{3} + 5 {x}^{2} - 4 x - 20$.

#### Explanation:

We know that: if $a$ is a zero of a real polynomial in $x$ (say), then $x - a$ is the factor of the polynomial.

Let $P \left(x\right)$ be the required polynomial.

Here $- 5 , 2 , - 2$ are the zeros of required polynomial.
$\implies \left\{x - \left(- 5\right)\right\} , \left(x - 2\right)$ and $\left\{x - \left(- 2\right)\right\}$ are the factors of the required polynomial.
$\implies P \left(x\right) = \left(x + 5\right) \left(x - 2\right) \left(x + 2\right) = \left(x + 5\right) \left({x}^{2} - 4\right)$
$\implies P \left(x\right) = {x}^{3} + 5 {x}^{2} - 4 x - 20$

Hence, the required polynomial is $P \left(x\right) = {x}^{3} + 5 {x}^{2} - 4 x - 20$