Univariate Polynomials

Definition: Univariate Polynomial

A univariate polynomial is a polynomial in a single variable.

NOTE

A univariate polynomial has the form

$$a_n{x}^n+\cdots a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$$

Properties

Theorem: Ring of Univariate Polynomials

Let $R$ be a commutative ring.

The set of all univariate polynomials over $R$ form a commutative ring together with polynomial addition and polynomial multiplication.

NOTATION

This ring is often denoted as $R[x]$ or $R[X]$.

PROOF

TODO

Equality

Theorem: Equality of Univariate Polynomials

Let $A(x)$ and $B(x)$ be two nonzero polynomials over an Integral Domain with degrees $\mathrm{deg}(A)$ and $\mathrm{deg}(B)$.

If $A(x)$ and $B(x)$ have the same value for $\max \{\mathrm{deg}(A),\mathrm{deg}(B)\}+1$ different values for $x$, then $A$ and $B$ are equal.

PROOF

TODO

Roots

Theorem: Roots of Univariate Polynomials

Let $A(x)$ be a univariate polynomial over a commutative ring $R$.

An element $p\in R$ is a root of $A$ if and only if $A$ is divisible by $x-p$.

PROOF

In the forward direction: Suppose that $p$ is a root of $A$, i.e. $A(p)=0$. The polynomial remainder theorem then tells us that the remainder of the division of $A$ by $x-p$ is zero, i.e. $A$ is divisible by $x-p$.

In the backward direction: Suppose that $A$ is divisible by $x-p$. This means that $A(x)=(x-p)Q(x)$ for some polynomial $Q$. Plugging in $p$ gives us $A(p)=(p-p)Q(p)=0\cdot Q(p)=0$ and so $p$ is a root of $A$.

Theorem: Number of Roots of Univariate Polynomials

If $P(x)$ is a polynomial $P(x)$ over an Integral Domain, then the maximum number of distinct roots which $P$ can have is equal its degree $\mathrm{deg}(P)$.

PROOF

TODO