# Find Zeros of Polynomial Functions - Problems

Find zeros of polynomial functions. Problems with detailled solutions are presented.

 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) = x4 - 2·x3 - 6·x2 + 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)] = x2 -(2 + i)x -(2 - i)x + (2+i)(2-i) = x2 - 4·x + 5 q(x) can be found by dividing p(x) by x2 - 4·x + 5. (x4 - 2·x3 - 6·x2 + 22·x - 15) / (x2 - 4·x + 5) = x2 + 2·x - 3 We now write p(x) in factored form p(x) = [x - (2 + i)][x - (2 - i)](x2 + 2·x - 3) The remaining 2 zeros of p(x) are the solutions to the quadratic equation. x2 + 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) = x4 + 6·x3 + 11·x2 + 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.