# 2.5 | Zeros of Polynomial Functions

The Fundamental Theorem of Algebra If $$f(x)$$ is a polynomial of degree $$n,$$ where $$n >0$$, then $$f$$ has at least one zero in the complex number system.
Linear Factorization Theorem If $$f(x)$$ is a polynomial of degree $$n,$$ where $$n >0,$$ then $$f$$ has precisely $$n$$ linear factors
$$f(x) = a_n(x -c_1)(x-c_2) \cdots (x-c_n)$$
where $$c_1, c_2, \ldots, c_n$$ are complex numbers.
The Rational Root Test
If the polynomial
$$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$$
has integer coefficients, every rational zero of $$f$$ has the form
$$\text{Rational zero} = \frac{p}{q}$$
where $$p$$ and $$q$$ have no common factors other than $$1$$, and

• $$p =$$ a factor of a constant term $$a_0$$
• $$q =$$ a factor of the leading coefficient $$a_n$$
Let $$f(x)$$ be a polynomial that has real coefficients. If $$a+bi,$$ where $$b \not = 0,$$ is a zero fo the function, the conjugate $$a-bi$$ is also a zero of the function.
Every polynomial of degree $$n>0$$ with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.
Decartes’s Rule of Signs Let
$$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$$
be a polynomial with real coefficients and $$a_0 \not = 0.$$

1. The number of positive real zeros of $$f$$ is either equal to the number of variations in sign of $$f(x)$$ or less than the number by an even integer.
2. The number of negative real zeros of $$f$$ is either equal to the number of variations in sign of $$f(-x)$$ or less than that number by an even integer.
Let $$f(x)$$ be a polynomial with real coefficients and a positive leading coefficient. Suppose $$f(x)$$ is divided by $$x-c,$$ using synthetic division.

1. If $$c > 0$$ and each number in the last row is either positive or zero, $$c$$ is an upper bound for the real zero of $$f$$.
2. If $$c < 0$$ and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), $$c$$ is a lower bound for the real zeros of $$f.$$