Polynomials

A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents combined using addition, subtraction, and multiplication.

• General Form: $f\left(x\right)=anx^{n}+an-x^{n-1}+\cdots+ax+a0$

• Where $an\ne0$

• Degree: The highest exponent n of the variable in the polynomial.

• Leading Coefficient: The coefficient an​ associated with the highest-degree term.

• Constant Term: The term a0​ with no variable attached.

Factor Theorem

The Factor Theorem links the roots of a polynomial to its linear factors and serves as a direct consequence of the Remainder Theorem.

• Statement: A linear binomial x−cx−c is a factor of a polynomial f(x)f(x) if and only if f(c)=0f(c)=0.

• Roots and Factors: If f(c)=0f(c)=0, then cc is a zero or root of f(x)f(x), meaning (x−c)(x−c) divides f(x)f(x) completely without a remainder.

Remainder Theorem

The Remainder Theorem provides a method to find the remainder when a polynomial f(x)f(x) is divided by a linear expression of the form x−cx−c.

• Statement: When a polynomial f(x)f(x) is divided by x−cx−c, the remainder is equal to f(c)f(c).

• Algebraic Representation: f(x)$f\left(x\right)\left(x-c\right)Q\left(x\right)+R$ where Q(x) is the quotient polynomial and R=f(c)R=f(c) is the constant remainder.