Expressions in Power: Polynomials
Polynomials are algebraic expressions made up of terms with whole-number exponents — from simple binomials like x + 3 to complex expressions like 4x³ - 2x² + 7x - 5. Adding, subtracting, multiplying, and factoring polynomials are essential skills for solving higher-degree equations, modeling curves, and building toward calculus.
Components of Polynomials
This section covers the key operations and vocabulary:
- Classifying Polynomials: By number of terms (monomial, binomial, trinomial) and by degree (the highest exponent present).
- Adding & Subtracting Polynomials: Combine like terms, keeping exponents and variables matched.
- Multiplying Polynomials: Use the distributive property (FOIL for binomials) to multiply each term by every term in the other polynomial.
- Factoring Polynomials: Reverse multiplication to express a polynomial as a product of simpler factors.
Examples of Polynomials
Classifying Examples
- 5x² is a monomial (one term) of degree 2.
- 3x + 7 is a binomial (two terms) of degree 1.
- 2x³ - x² + 4x - 9 is a polynomial with four terms and degree 3.
Adding & Subtracting Examples
- Add (3x² + 2x - 5) + (x² - 4x + 3): Combine like terms to get 4x² - 2x - 2.
- Subtract (5x² + x - 7) - (2x² - 3x + 1): Distribute the negative and combine: 3x² + 4x - 8.
- Add (4a³ - 2a) + (a³ + 5a - 3): Combine like terms to get 5a³ + 3a - 3.
Multiplying Examples
- Multiply (x + 3)(x + 5) using FOIL: x² + 5x + 3x + 15 = x² + 8x + 15.
- Multiply (2x - 1)(x + 4): 2x² + 8x - x - 4 = 2x² + 7x - 4.
- Multiply (x + 2)(x² - 3x + 1): Distribute each term: x³ - 3x² + x + 2x² - 6x + 2 = x³ - x² - 5x + 2.
Factoring Examples
- Factor x² + 5x + 6: Find two numbers that multiply to 6 and add to 5 — that's 2 and 3, giving (x + 2)(x + 3).
- Factor x² - 9: This is a difference of squares, so it factors as (x + 3)(x - 3).
- Factor 2x² + 7x + 3: Find factors of 2 × 3 = 6 that add to 7 — that's 6 and 1. Rewrite as 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).