2 – 3i is a zero of p(x) =
x^{3} – 3x^{2} + 9x + 13 as shown here:
p(2 – 3i) = (2 – 3i)^{3} – 3(2 – 3i)^{2} +
9(2 – 3i) + 13
= (–46 – 9i) – 3(–5 – 12i) + (18 – 27i) + 13
= –46 – 9i^{} + 15 + 36i + 18 – 27i + 13
= 0.
By the conjugate pair theorem, 2 + 3i is also
a zero of p(x).
p(2 + 3i) = (2 + 3i)^{3} – 3(2 + 3i)^{2} + 9(2 + 3i) + 13
= (–46 + 9i)^{} – 3(–5 + 12i)^{} + (18 + 27i) + 13
= –46 + 9i^{} + 15 – 36i + 18 + 27i + 13
=
0. |