Introduction to Polynomials

Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. They’re versatile tools in math, representing everything from simple lines to complex curves.

Definition

A polynomial looks like: \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \), where \(n\) is the degree (highest exponent), and \(a_n, a_{n-1}, \ldots, a_0\) are coefficients.

Example: \( 3x^2 + 2x - 1 \) (degree 2).

Types of Polynomials

• Monomial: One term, e.g., \( 5x^3 \).
• Binomial: Two terms, e.g., \( x^2 - 4 \).
• Trinomial: Three terms, e.g., \( x^2 + 3x + 2 \).

Graphing Polynomials

Consider \( y = x^3 - 2x \), a cubic polynomial:

The degree (3) indicates it can cross the x-axis up to 3 times.

Operations

Addition: \( (2x^2 + 3x) + (x^2 - x) = 3x^2 + 2x \).

Multiplication: \( (x + 2)(x - 3) = x^2 - x - 6 \).

Applications

Polynomials model motion (e.g., \( s(t) = -16t^2 + v_0 t + h_0 \) for falling objects) and financial growth.