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 - 2x3 - 6x2 + 22x - 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 - 4x + 5

  • q(x) can be found by dividing p(x) by x2 - 4x + 5.
    (x4 - 2x3 - 6x2 + 22x - 15) / (x2 - 4x + 5)

    = x2 + 2x - 3

  • We now write p(x) in factored form
    p(x) = [x - (2 + i)][x - (2 - i)](x2 + 2x - 3)

  • The remaining 2 zeros of p(x) are the solutions to the quadratic equation.
    x2 + 2x - 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 + 6x3 + 11x2 + 6x + 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.


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Updated: 2 April 2013

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