3.6 Zeros of polynomial functions  (Page 4/14)

Using the fundamental theorem of algebra

Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.

Suppose $f$ is a polynomial function of degree four, and $f(x)=0.$ The Fundamental Theorem of Algebra states that there is at least one complex solution, call it $c_1.$ By the Factor Theorem, we can write $f(x)$ as a product of $x-c_{\text{1}}$ and a polynomial quotient. Since $x-c_{\text{1}}$ is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it $c_{\text{2}}.$ So we can write the polynomial quotient as a product of $x-c_{\text{2}}$ and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of $f(x).$

Using the linear factorization theorem to find polynomials with given zeros

A vital implication of the Fundamental Theorem of Algebra    , as we stated above, is that a polynomial function of degree $n$ will have $n$ zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into $n$ factors. The Linear Factorization Theorem    tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form $(x-c),$ where $c$ is a complex number.