Operations with Polynomials

Addition and Subtraction

Combine like terms. For P(x) = 3x² − 2x + 1 and Q(x) = x² + 5x − 3: P + Q = 4x² + 3x − 2; P − Q = 2x² − 7x + 4.

Multiplication

Use the distributive property. For a monomial times a polynomial: 2x(3x² − x + 4) = 6x³ − 2x² + 8x.

For two polynomials: (x + 2)(x − 3) = x² − x − 6.

Special Product Formulas

Difference of squares: (a + b)(a − b) = a² − b²

Square of a sum: (a + b)² = a² + 2ab + b²

Square of a difference: (a − b)² = a² − 2ab + b²

Sum of cubes: (a + b)(a² − ab + b²) = a³ + b³

Difference of cubes: (a − b)(a² + ab + b²) = a³ − b³

Division

Long division of polynomials follows the same logic as numeric long division. The Remainder Theorem: P(x) divided by (x − a) gives remainder P(a). The Factor Theorem: (x − a) is a factor of P(x) if and only if P(a) = 0.