Polynomials Remainder And Factor Theorems

Polynomials are continuous, smooth functions defined everywhere on $ℝ$ . The Remainder and Factor Theorems provide powerful tools for factoring polynomials and integrating rational functions via partial fraction decomposition.

Quick Reference

Theorem / Rule	Algebraic Form	Description / Significance
Polynomial Form	$P (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$	Degree n ( $a_{n} \neq 0$ )
Division Algorithm	$P (x) = D (x) Q (x) + R (x)$	$deg (R) < deg (D)$
Remainder Theorem	$R = P (c)$	Remainder when dividing $P (x)$ by $x - c$
Factor Theorem	$P (c) = 0 ⟺ (x - c) factors P (x)$	Link between polynomial roots and linear factors

Polynomial Definition and Degree

A polynomial of degree $n$ is a function of the form:

P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0} (a_{n} \neq 0)

where $n \geq 0$ is an integer and $a_{n}, a_{n - 1}, \dots, a_{0}$ are real numbers called coefficients.

Division Algorithm for Polynomials

If a polynomial $P (x)$ is divided by a non-zero divisor polynomial $D (x)$ , there exist unique polynomials $Q (x)$ (quotient) and $R (x)$ (remainder) such that:

P (x) = D (x) Q (x) + R (x)

where either $R (x) = 0$ or the degree of $R (x)$ is strictly less than the degree of $D (x)$ .

The Remainder Theorem

Remainder Theorem: If a polynomial $P (x)$ is divided by $x - c$ , then the remainder is equal to $P (c)$ .

Find the remainder when $P (x) = 3 x^{3} - 4 x^{2} + 2 x - 5$ is divided by $x - 2$ .

Solution

By the Remainder Theorem, the remainder $R$ is equal to $P (2)$ :

\begin{aligned} P(2) &= 3(2)^3 - 4(2)^2 + 2(2) - 5 \\ &= 3(8) - 4(4) + 4 - 5 \\ &= 24 - 16 + 4 - 5 = 7 \end{aligned}

The remainder is 7.

The Factor Theorem

Factor Theorem: A polynomial $P (x)$ has a factor $(x - c)$ if and only if $P (c) = 0$ .

Show that $(x + 1)$ is a factor of $P (x) = x^{3} + 4 x^{2} + x - 2$ .

Solution

Set $x + 1 = 0 ⟹ c = - 1$ . Evaluate $P (- 1)$ :

\begin{aligned} P(-1) &= (-1)^3 + 4(-1)^2 + (-1) - 2 \\ &= -1 + 4(1) - 1 - 2 \\ &= -1 + 4 - 1 - 2 = 0 \end{aligned}

Since $P (- 1) = 0$ , by the Factor Theorem, $(x + 1)$ is a factor of $P (x)$ .

Quick Reference

Polynomial Definition and Degree

Division Algorithm for Polynomials

The Remainder Theorem

The Factor Theorem

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