Algebra 101 - Unit 5

Lesson 24: Introduction to Polynomials

A polynomial is an algebraic expression made up of one or more terms, where each term consists of a coefficient and a variable raised to a non-negative integer exponent.

General form of a polynomial in one variable x:

a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀, where:

• a_n, a_n-1, …, a₀ are coefficients
• n is a non-negative integer (degree of the polynomial)

Examples of Polynomials

• x² + 3x + 2 (quadratic polynomial, degree 2)
• 4x³ - x + 7 (cubic polynomial, degree 3)
• 5 (constant polynomial, degree 0)
• 2x⁵ - 3x² + x - 1 (degree 5)

Non-examples (not polynomials):

• 1/x or x⁻¹ (negative exponent)
• √x or x^(1/2) (fractional exponent)
• sin(x), e^x (functions, not polynomials)

Terms and Degree

• Term: A single part of a polynomial, separated by + or −. Example: in 3x² + 5x − 7, the terms are 3x², 5x, and −7.
• Degree: The highest exponent of the variable. Example: in 4x³ - x + 7, the degree is 3.
• Leading Coefficient: The coefficient of the term with the highest degree. Example: in 4x³ - x + 7, the leading coefficient is 4.
• Constant Term: The term without a variable. Example: in 4x³ - x + 7, the constant term is 7.

Practice Problems

1. Identify the degree and leading coefficient: 2x⁴ + 3x² − x + 5
2. Classify each expression as a polynomial or not: x² − √x + 4
3. Write the terms of: 5x³ − 2x + 9
4. Identify the constant term: 7x² − 3x + 11