Polynomials

What is a Polynomial?

A polynomial is an expression made of terms with non-negative integer exponents:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀

Examples: 4x³ - 2x² + 7, x + 5, 3x² - x

Not polynomials: x⁻¹, √x, 1/x (these involve negative or fractional powers)

Key Vocabulary

• Degree: the highest power of x → 4x³ - 2x² + 7 has degree 3
• Coefficient: the number in front of a term → in 4x³, the coefficient is 4
• Constant term: the term with no variable → in 5x³ - 3x + 8, it's 8
• Leading coefficient: coefficient of the highest-degree term

Adding and Subtracting

Collect like terms (same variable and power):

• (3x² + 2x - 1) + (x² - 5x + 4) = (3+1)x² + (2-5)x + (-1+4) = 4x² - 3x + 3

Multiplying Polynomials

Use the FOIL method for two binomials, or multiply each term in one bracket by each term in the other:

(x + 3)(x - 2):

• x × x = x²
• x × (-2) = -2x
• 3 × x = 3x
• 3 × (-2) = -6

Result: x² - 2x + 3x - 6 = x² + x - 6

Special Products

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b² (difference of squares)