Polynomial Multiplication

Multiply two polynomials — example and distributive law

Polynomial Multiplication Calculator

What is calculated?

This function computes the multiplication of two polynomials using the distributive law. Each term of the first polynomial is multiplied by every term of the second polynomial.

Enter polynomials
Input format

Coefficients can be entered as a sequence of numbers separated by semicolons or spaces.

Example: For \(3x^2+4x+5\) enter "3 4 5" or "3; 4; 5"

Distributive law

Each term of the first polynomial is multiplied by every term of the second: (a+b)×(c+d) = ac + ad + bc + bd


×
Result
P(x) =
Q(x) =
Product =
Each term multiplied by every term, then like terms are collected

Multiplication info

Distributive law

Multiply each by each:

  • Expand the first polynomial
  • Each term × every term
  • Add exponents: x^a × x^b = x^(a+b)
  • Collect like terms

Rule of thumb: When multiplying, exponents add and coefficients multiply.

Quick examples
x × x² = x³
Exponents add
3x × 2x = 6x²
Coefficients multiply
5 × x = 5x
Constant times variable
(a+b)(c+d) = ac+ad+bc+bd
Distributive
FOIL method

First, Outer, Inner, Last - systematic approach for binomial multiplication.


Formulas for polynomial multiplication

Distributive law
\[P(x) \cdot Q(x) = \sum_{i,j} a_i b_j x^{i+j}\] General multiplication
Binomial multiplication
\[(a+b)(c+d) = ac + ad + bc + bd\] FOIL method
Power rules
\[x^a \cdot x^b = x^{a+b}\] Exponents add
Degree of product
\[\deg(P \cdot Q) = \deg(P) + \deg(Q)\] Degrees add

Step-by-step example

Example: (3x² + 4x + 5) × (2x + 3)

1Write polynomials

\[P(x) = 3x^2 + 4x + 5\] \[Q(x) = 2x + 3\]

Given polynomials

2Apply distributive law

(3x² + 4x + 5) × (2x + 3)

Multiply each term by each

Multiplication grid
2x
3
Products
3x²
6x³
9x²
→ 6x³ + 9x²
4x
8x²
12x
→ 8x² + 12x
5
10x
15
→ 10x + 15

3Write all terms

\[6x^3 + 9x^2 + 8x^2 + 12x + 10x + 15\]

All individual products

4Collect like terms

x³: 6x³
x²: 9x² + 8x² = 17x²
x¹: 12x + 10x = 22x
x⁰: 15

5Final product

\[P(x) \times Q(x) = 6x^3 + 17x^2 + 22x + 15\]

Final product

FOIL method for binomials

Special case: (a + b)(c + d)
FOIL procedure
  • First: Multiply first terms → ac
  • Outer: Multiply outer terms → ad
  • Inner: Multiply inner terms → bc
  • Last: Multiply last terms → bd
Example: (x + 2)(x + 3)
F: x × x = x²
O: x × 3 = 3x
I: 2 × x = 2x
L: 2 × 3 = 6
= x² + 5x + 6




Polynomial Functions

Add  •  Scalar Divide  •  Divide Pointwise  •  Divide with Remainder  •  Multiply  •  Pointwise Multiply  •  Scalar Multiply  •  Subtract  •