Polynomial Multiplication Calculator

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Reviewed by: David Chen, CFA | Mathematics Educator & Quantitative Analyst

Multiply any two polynomials instantly. This calculator handles various degrees and variables, providing a step-by-step expansion using the distributive property.

Polynomial Multiplication Calculator

Invalid format. Use 3x^2 + 2x + 1
Invalid format. Use x – 3
Resultant Polynomial
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Polynomial Multiplication Calculator Formula

P(x) \times Q(x) = \sum_{i=0}^{n} a_i x^i \times \sum_{j=0}^{m} b_j x^j = \sum_{k=0}^{n+m} c_k x^k

Where $c_k = \sum_{i+j=k} a_i b_j$. This is the mathematical representation of the distributive property (FOIL for binomials).

Source: Wolfram MathWorld – Polynomial Multiplication

Variables:

  • Polynomial A ($P(x)$): The first expression consisting of variables and coefficients.
  • Polynomial B ($Q(x)$): The second expression to be multiplied.
  • Degree ($n, m$): The highest power of the variable in each polynomial.
  • Coefficients ($a_i, b_j$): The numerical values multiplying the variables.

What is Polynomial Multiplication?

Polynomial multiplication is a fundamental algebraic operation where two polynomials are multiplied together to form a new polynomial. The process involves multiplying every term of the first polynomial by every term of the second polynomial and then combining like terms.

This operation is essential in fields ranging from engineering and physics to computer science (cryptography and signal processing). Understanding how to expand these expressions allows for deeper analysis of functions and their roots.

How to Calculate Polynomial Multiplication (Example)

Example: Multiply $(2x + 3)$ and $(x – 5)$

  1. Distribute the first term: $2x \times x = 2x^2$ and $2x \times (-5) = -10x$.
  2. Distribute the second term: $3 \times x = 3x$ and $3 \times (-5) = -15$.
  3. Combine terms: $2x^2 – 10x + 3x – 15$.
  4. Final Result: $2x^2 – 7x – 15$.

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Frequently Asked Questions (FAQ)

What is the FOIL method? FOIL stands for First, Outer, Inner, Last. It is a specific shortcut for multiplying two binomials (polynomials with two terms).

Can I multiply polynomials with different variables? Yes, the distributive property still applies, but you cannot combine terms unless both variables and their powers match perfectly.

What happens to the degrees when multiplying? The degree of the resulting polynomial is always the sum of the degrees of the two input polynomials.

Is polynomial multiplication commutative? Yes, $A(x) \times B(x) = B(x) \times A(x)$, meaning the order of multiplication does not change the result.

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