Learn/Algebra/Polynomials in One Variable

Algebra • Topic 14

Polynomials in One Variable

Polynomials are central to Olympiad algebra. We treat them as formal algebraic objects, focusing on properties like degree, divisibility, and coefficients.

Definition

Definition (Polynomial)

A polynomial

P (x)

over a field (usually

R

C

) is an expression:

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

where

a_{n} \neq = 0

. The integer

n

is the degree, denoted

de g P

Division Algorithm

Theorem (Polynomial Division)

For any polynomials

P (x)

and

D (x)

with

D (x) \neq = 0

, 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

de g R < de g D

Remainder Theorem

Theorem (Remainder Theorem)

The remainder when

P (x)

is divided by

(x - c)

is equal to

P (c)

Proof. By the division algorithm: $P (x) = (x - c) Q (x) + R$ . Since the divisor is degree 1, $R$ is a constant. Substitute $x = c$ : $P (c) = (c - c) Q (c) + R = 0 + R = R$ . ∎

Factor Theorem

Theorem (Factor Theorem)

(x - c)

is a factor of

P (x)

if and only if

P (c) = 0

Practice Problems

Exercise (Problem 1)

Find the remainder when

x^{2023} + 1

is divided by

x - 1

Exercise (Problem 2)

Determine the polynomial

P (x)

of degree 2 such that

P (0) = 1

P (1) = 2

, and

P (2) = 5

Exercise (Problem 3)

P (x)

leaves a remainder of 3 when divided by

x - 1

and a remainder of 5 when divided by

x - 3

, find the remainder when

P (x)

is divided by

(x - 1) (x - 3)