Question

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

Answer 1

Answer:

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

Step-by-step 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 $f(x)=x^4+15x^2-16$ .