Polynomial Multiplication Calculator – Multiply Polynomials
Multiply polynomials and see step-by-step solutions
Table of Contents
How to Use
- Enter coefficients for the first polynomial (up to degree 2)
- Enter coefficients for the second polynomial (up to degree 2)
- Leave coefficients as 0 for missing terms
- Click calculate to see the product
- Review the step-by-step multiplication process
What is Polynomial Multiplication?
Polynomial multiplication is the process of multiplying two polynomials together to produce a new polynomial. It uses the distributive property, where each term of the first polynomial is multiplied by each term of the second polynomial, and like terms are combined.
The degree of the resulting polynomial equals the sum of the degrees of the two original polynomials.
The FOIL Method
For multiplying two binomials (polynomials with two terms), the FOIL method is commonly used:
- F - First: Multiply the first terms of each binomial
- O - Outer: Multiply the outer terms
- I - Inner: Multiply the inner terms
- L - Last: Multiply the last terms of each binomial
- Combine like terms to get the final result
Example
Multiply (2x + 3) by (x - 4):
- First: 2x × x = 2x²
- Outer: 2x × (-4) = -8x
- Inner: 3 × x = 3x
- Last: 3 × (-4) = -12
- Combine: 2x² + (-8x) + 3x + (-12) = 2x² - 5x - 12
Special Products
Some polynomial multiplications follow special patterns:
- Square of a sum: (a + b)² = a² + 2ab + b²
- Square of a difference: (a - b)² = a² - 2ab + b²
- Difference of squares: (a + b)(a - b) = a² - b²
- Sum of cubes: (a + b)(a² - ab + b²) = a³ + b³
- Difference of cubes: (a - b)(a² + ab + b²) = a³ - b³
Applications of Polynomial Multiplication
- Expanding algebraic expressions
- Finding areas and volumes in geometry
- Physics calculations involving motion and forces
- Economics and business modeling
- Signal processing and filter design
- Computer graphics and animation
- Cryptography and coding theory
Frequently Asked Questions
- What is the degree of the product of two polynomials?
- The degree of the product equals the sum of the degrees of the two polynomials being multiplied. For example, multiplying a degree 2 polynomial by a degree 3 polynomial gives a degree 5 polynomial.
- Can I multiply polynomials with more than two terms?
- Yes, the distributive property works for polynomials of any size. Each term of the first polynomial is multiplied by each term of the second polynomial, then like terms are combined.
- What happens when I multiply a polynomial by a constant?
- When multiplying by a constant (a polynomial of degree 0), simply multiply each coefficient of the polynomial by that constant. The degree of the result stays the same.
- How do I handle missing terms in a polynomial?
- For missing terms, use a coefficient of 0. For example, x² + 5 (missing the x term) would have coefficients: 1, 0, 5.