3.6 Zeros of polynomial functions  (Page 5/14)

Let $f$ be a polynomial function with real coefficients, and suppose $a+bi\text{, }b\ne 0,$ is a zero of $f(x).$ Then, by the Factor Theorem, $x-(a+bi)$ is a factor of $f(x).$ For $f$ to have real coefficients, $x-(a-bi)$ must also be a factor of $f(x).$ This is true because any factor other than $x-(a-bi),$ when multiplied by $x-(a+bi),$ will leave imaginary components in the product. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function $f$ with real coefficients has a complex zero $a+bi,$ then the complex conjugate $a-bi$ must also be a zero of $f(x).$ This is called the Complex Conjugate Theorem .

Using descartes’ rule of signs

There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order, Descartes’ Rule of Signs tells us of a relationship between the number of sign changes in $f(x)$ and the number of positive real zeros. For example, the polynomial function below has one sign change.

This tells us that the function must have 1 positive real zero.