Question

Write a polynomial function f of the least degree that has rational coefficients,

a leading coefficient of 1, and the given zeros: -1, 1, -4i, 4i, in standard form.

Answer 1

`f(x)=x^4+15x^2-16`

Explanation:

It is given that the leading coefficient of the polynomial is 1.

We need to find a polynomial function f of the least degree.

If c is a root of a polynomial then (x-c) is a factor of the polynomial.

The given zeros are -1, 1, -4i, 4i. So,

`f(x)=(x-1)(x+1)(x+4i)(x-4i)`

`f(x)=(x^2-1^2)(x^2-(4i)^2)`

`f(x)=(x^2-1)(x^2+16)`
`[\because i^2=-1]`

`f(x)=x^2(x^2+16)-1(x^2+16)`

`f(x)=x^4+16x^2-x^2-16`

`f(x)=x^4+15x^2-16`

Therefore, the required polynomial is
`f(x)=x^4+15x^2-16` .