Polynomials, Remainder and Factor Theorems

Concept	Statement
Degree	Highest exponent of $x$ with nonzero coefficient
Division algorithm	$P(x)=D(x)Q(x)+R(x)$,
Remainder theorem	Remainder when $P(x)÷(x-c)$ is $P(c)$
Factor theorem	$P(c)=0⟺(x-c)$ is a factor of $P(x)$

Polynomial Functions

A polynomial function of degree $n\ge 0$ has the form

$$P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0,$$

where $a_n,\dots ,a_0$ are real numbers called the coefficients. The term $a_n{x}^n$ (with $a_n\ne 0$) is the leading term, and $n$ is the degree.

For example, $P(x)=3x^4-\sqrt{2}{x}^2-\frac{1}{8}x+\pi$ is a polynomial of degree 4.

Polynomial Division

If $P(x)$ and $D(x)$ are polynomials with $\mathrm{deg}(D)\le \mathrm{deg}(P)$, then there exist unique polynomials $Q(x)$ (the quotient) and $R(x)$ (the remainder) such that

• $P(x)$ is called the dividend and $D(x)$ the divisor.
• If $R(x)\equiv 0$, then $P(x)$ is divisible by $D(x)$, and $D(x)$ is a factor of $P(x)$.

The Remainder Theorem

When we divide $P(x)$ by $x-c$, the divisor has degree 1, so the remainder must have degree less than 1, meaning it is just a constant $r$:

$$P(x)=(x-c)Q(x)+r.$$

Setting $x=c$:

$$P(c)=\underset{0}{\underbrace{(c-c)}}Q(c)+r⟹r=P(c).$$

The Factor Theorem

If $P(c)=0$, then the remainder is zero, so $P(x)=(x-c)Q(x)$, which means $x-c$ is a factor of $P(x)$. Conversely, if $x-c$ is a factor of $P(x)$, then $r=0$ and $P(c)=0$.

Example. Consider $P(x)=2x^3+x^2-4x+4$.

• Since $P(1)=2+1-4+4=3$, the remainder when $P(x)$ is divided by $x-1$ is 3. Indeed: $$P(x)=(x-1)(2x^2+3x-1)+3.$$
• Since $P(-2)=2(-8)+4+8+4=-16+16=0$, the factor theorem tells us $x+2$ is a factor. Indeed: $$P(x)=(x+2)(2x^2-3x+2).$$

Frequently Asked Questions

How do I perform polynomial long division?

Polynomial long division mirrors numerical long division. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply the divisor by this term, subtract from the dividend, and repeat with the remainder. Continue until the degree of the remainder is less than the degree of the divisor. For a step-by-step walkthrough, see the long division supplement or the external link in the footnote of the source text.

What does it mean for $c$ to be a "root" of a polynomial?

A **root** (or **zero**) of $P(x)$ is a number $c$ such that $P(c)=0$. By the factor theorem, $c$ is a root if and only if $(x-c)$ is a factor. A polynomial of degree $n$ has at most $n$ roots (counting multiplicity) by the fundamental theorem of algebra.

How are polynomials used in calculus beyond the factor theorem?

Polynomials are used in many ways: as approximations to more complicated functions (Taylor and Maclaurin polynomials), as the standard test cases for differentiation and integration rules, and in partial fraction decomposition (which reduces rational functions into sums of simpler fractions before integration). The factor theorem is specifically useful for simplifying rational functions that arise in limit computations.