Question

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?

Answer 1

The required polynomial is `P(x)=x^3+5x^2-4x-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(x)` be the required polynomial.

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

Hence, the required polynomial is `P(x)=x^3+5x^2-4x-20`