Suppose p (x) is a polynomial with real coefficients and p (4 + 9i)=0. What is p(4-9i)?
It is also zero. Complex roots of polynomials (if they exist at all) occur in conjugate pairs, that is,
a + ib and a - ib.
To find the value of p(4-9i), we can use the complex conjugate theorem. The complex conjugate theorem states that if a polynomial with real coefficients has a complex number a+bi as a root, then its conjugate a-bi is also a root.
Given that p(4+9i) = 0, we can conclude that p(4-9i) = 0 as well. This is because the complex conjugate of 4+9i is 4-9i.
Therefore, p(4-9i) = 0.