6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.

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6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor of the polynomial function f(x). c is a root or solution of the polynomial equation f(x) = 0

Fundamental Theorem of Algebra Since real numbers and imaginary numbers both belong to the set of complex numbers, all polynomial equations with degree greater than zero will have at least one root in the set of complex numbers. A polynomial function cannot have more real zeros than its degree. Remember – zeros are where the graph crosses the x-axis.

Determine the number and type of roots Solve each equation and state the number and type of roots. One Real Repeated Root Two Real Roots Two Imaginary Roots One Real Root Two Imaginary Roots

Corollary A polynomial equation of the form P(x) = 0 of degree n with complex coefficients has exactly n roots in the set of complex numbers. Similarly, a polynomial function of nth degree has exactly n zeros.

Descartes’ Rule of Signs In order to find the number of positive real zeros of a polynomial function written in descending order: Count the number of changes in sign of the coefficients of the terms. It will be this number or less than this by an even number (subtract 2). f(x) = x3 - 11x2 + 40x - 50 Positive real zeros would be 3 or 1. In order to find the number of negative real zeros, you must substitute (-x) in for the x, then count the same as before. No negative reals. f(x) = -x3 - 11x2 - 40x - 50

Example Discuss the real zeros of 3 or 1 Positive Real Zeros 3 or 1 Negative Real Zeros

Conjugates If z = a + bi is a complex number and is also a zero of a polynomial function, then its conjugate is also a zero. Example: 3 – 4i -------- conjugate = 3 + 4i Example: 2 + 3i -------- conjugate = 2 – 3i\

Example Write a polynomial function of least degree with integral coefficients the zeros of which include –1, and 1 + 2i. Write the polynomial as a product of its factors. Multiply the factors using distribution.

Homework Assignment #38 p. 366 11-37 odd, 42-43, 51-52