Review
Note: In what follows the imaginary unit i is defined as i = square root (1)
Let p(x) be a polynomial function with real coefficients. If a + ib is an imaginary zero of p(x), the conjuagte a  bi is also a zero of p(x).
TUTORIAL
Example  Problem 1: 2 + i is a zero of polynomial p(x) given below, find all the other zeros.
p(x) = x^{4}  2·x^{3}  6·x^{2} + 22·x  15
Solution to Problem 1:
 The zero 2 + i is an imaginary number and p(x) has real coefficients. It follows that the conjugate 2  i is also a zero of p(x). p(x) may be written in factored form as follows
p(x) = [x  (2 + i)][x  (2  i)]q(x)
 Let us expand the term [x  (2 + i)][x  (2  i)] in p(x)
[x  (2 + i)][x  (2  i)] = x^{2} (2 + i)x (2  i)x + (2+i)(2i)
= x^{2}  4·x + 5
 q(x) can be found by dividing p(x) by x^{2}  4·x + 5.
(x^{4}  2·x^{3}  6·x^{2} + 22·x  15) / (x^{2}  4·x + 5)
= x^{2} + 2·x  3
 We now write p(x) in factored form
p(x) = [x  (2 + i)][x  (2  i)](x^{2} + 2·x  3)
 The remaining 2 zeros of p(x) are the solutions to the quadratic equation.
x^{2} + 2·x  3 = 0
 Factor the above quadratic equation and solve.
(x  1)·(x + 3) = 0
solutions
x = 1
x = 3
 p(x) has the following zeros.
2 + i , 2  i, 3 and 1.
Matched Problem 1: 3  i is a zero of polynomial p(x) given below, find all the other zeros.
p(x) = x^{4} + 6·x^{3} + 11·x^{2} + 6·x + 10
More references and links to polynomial functions.
Factor Polynomials.
Polynomial Functions  Interactive Tutorial Using Applet.
Polynomial Functions in Factored Form.
Polynomial Functions, Zeros, Factors and Intercepts
Graphs of Polynomial Functions  Self Test.
