Question

What is the relationship between the degree of a polynomial and the maximum number of zeros it can have?

Answer 1

The relationship between the degree of a polynomial and the maximum number of zeros.

According to the fundamental theorem of algebra, the maximum number of zero's is equal to the degree of polynomials.

Let $\mathrm{P}\left(\mathrm{x}\right)$ be the polynomial with a degree $\text{'}\mathrm{n}\text{'}$.

A polynomial degree $\text{'}\mathrm{n}\text{'}$ is a function of the form:
$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{a}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}+{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{x}}^{\mathrm{n}-1}+\dots +{\mathrm{a}}_1\mathrm{x}+{\mathrm{a}}_0$. Where ${\mathrm{a}}_{\mathrm{n}}\ne 0$.

The numbers ${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2,\dots ,{\mathrm{a}}_{\mathrm{n}}$ are called the coefficients of the polynomial.

The $a_0$ is the constant coefficient or constant term.

The number ${\mathrm{a}}_{\mathrm{n}}$, the coefficient of the highest power, is the leading coefficient, and the term ${\mathrm{a}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}$ is the leading term.

$\begin{array}{rcl}& \Rightarrow & \mathrm{degree}\mathrm{P}\left(\mathrm{x}\right)=\mathrm{n}\\ & \Rightarrow & \mathrm{maximum}\mathrm{number}\mathrm{of}\mathrm{zero}\text{'}\mathrm{s}=\mathrm{n}\end{array}$

Hence, the relationship between the degree of a polynomial and the maximum number of zeros is polynomial with the degree $\mathbf{n}$ can have at most $\mathbf{n}$ zeros.