How to Multiply Polynomials
Multiplying polynomials involves distributing each term of one polynomial to every term of the other polynomial, then combining like terms. This process is sometimes called the FOIL method for binomials, but the distributive property applies to polynomials of any degree.
Methods of Polynomial Multiplication
FOIL Method
For two binomials: multiply First, Outer, Inner, Last terms.
(a+b)(c+d) = ac + ad + bc + bd
Distributive Property
Multiply each term in the first polynomial by each term in the second.
Each term x Each term, then combine
Box Method
Arrange terms in a grid and multiply to fill each cell.
Grid layout for visual organization
Step-by-Step Process
- Distribute: Take each term from the first polynomial and multiply it by every term in the second polynomial.
- Multiply terms: Multiply coefficients together and add exponents of like variables.
- Combine like terms: Group terms with the same degree and add their coefficients.
- Order: Write the result in standard form (descending order of degree).
Example
Multiply (2x + 3) by (x² - x + 4):
- 2x times x² = 2x³
- 2x times (-x) = -2x²
- 2x times 4 = 8x
- 3 times x² = 3x²
- 3 times (-x) = -3x
- 3 times 4 = 12
- Combine: 2x³ + x² + 5x + 12
Special Products
- Difference of Squares: (a + b)(a - b) = a² - b²
- Perfect Square Trinomial: (a + b)² = a² + 2ab + b²
- Sum of Cubes: (a + b)(a² - ab + b²) = a³ + b³
Applications
Polynomial multiplication is essential in algebra, calculus, physics, and engineering. It is used in signal processing, computer graphics, error-correcting codes, and many areas of applied mathematics.