Question

If a polynomial function, f(x), with rational coefficients has roots 3 and startroot 7 endroot, what must also be a root of f(x)?

1. minhkhang
The conjugate which is 3 – √7 is also a root of f(x).

### What is the root of a polynomial function?

The root of a polynomial function f(x) is the value of x for which f(x) = 0.
Now if a polynomial function has a root x = a + √b then the conjugate of x which is x’ = a – √b is also a root of the function, f(x).

### What must also be a root of f(x)?

Given that the polynomial function, f(x), with rational coefficients has roots
3 + √7, then by the above, the conjugate which is 3 – √7 is also a root of f(x).
So, 3 – √7 is also a root of f(x).