Introduction
A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Key Terms
- Monomial: Polynomial with 1 term.
- Binomial: Polynomial with 2 terms.
- Trinomial: Polynomial with 3 terms.
Remainder Theorem
Let p(x) be any polynomial of degree greater than or equal to 1 and let a be any real number. If p(x) is divided by the linear polynomial x - a, then the remainder is p(a).
Example:
Find the remainder when x⁴ + x³ - 2x² + x + 1 is divided by x - 1.
Solution: p(1) = 1 + 1 - 2 + 1 + 1 = 2.
Factor Theorem
If p(a) = 0, then x - a is a factor of p(x).
This is extremely useful for factorizing cubic polynomials where splitting the middle term doesn't work directly.