Answer:
If 5 + 6i is a root of the polynomial function f(x), then its complex conjugate 5 - 6i must also be a root of f(x). This is because complex roots of polynomial functions always come in conjugate pairs.
To see why this is true, consider a polynomial function with real coefficients. If a complex number z = a + bi is a root of the polynomial, then we have:
f(z) = 0
Substituting z = a + bi into the polynomial function, we get:
f(a + bi) = 0
Now we can take the complex conjugate of both sides:
f(a - bi) = (f(a + bi))^*
Since the coefficients of the polynomial are real, we have:
(f(a + bi))^* = f(a - bi)
Therefore, if a + bi is a root of the polynomial, then so is its conjugate a - bi.
In this case, since 5 + 6i is a root of f(x), we know that 5 - 6i must also be a root of f(x). Therefore, the answer is the complex number 5 - 6i.