Polynomials and Factoring

Lesson 9 of 10Grades 8–9

A polynomial is a sum of terms with variables raised to whole-number exponents. Adding, subtracting, and multiplying polynomials follows the same rules as combining like terms and distributing. Factoring is the reverse — rewriting as a product.

Key Concepts

Adding and Multiplying Polynomials

To add: combine like terms. (3x² + 2x) + (x² − 5x + 1) = 4x² − 3x + 1. To multiply: use FOIL for binomials. (x+3)(x−2) = x² − 2x + 3x − 6 = x² + x − 6. FOIL = First, Outer, Inner, Last terms.

Factoring Out the GCF

Always factor out the greatest common factor first. 6x² + 9x = 3x(2x + 3). The GCF of 6x² and 9x is 3x. Check: 3x(2x+3) = 6x²+9x ✓.

Factoring Trinomials

x² + bx + c: find two numbers that multiply to c and add to b. x² + 5x + 6: 2 × 3 = 6, 2 + 3 = 5 → (x+2)(x+3). x² + bx − c: one number is negative. x² + 2x − 8: −2 × 4 = −8, −2 + 4 = 2 → (x−2)(x+4).

🆕 Polynomial Multiplier (FOIL)

Multiply two binomials step by step. Enter (x+a)(x+b).

(x + )(x + )

✅ Check Your Understanding

1. FOIL stands for...

2. Factor: 4x² + 8x

3. Factor x² + 7x + 12. The factors are...