Polynomial Pointwise Multiplication

Calculator and formulas for pointwise multiplication of two polynomials

Pointwise Multiplication Calculator

What is calculated?

This function computes the pointwise multiplication of two polynomials. Corresponding coefficients of like powers are multiplied together.

Enter polynomials

For 3x² + 4x + 5 enter: 3 4 5

×

For 2x + 3 enter: 2 3
Result
P(x):
Q(x):
Product:
The result displays the pointwise multiplication of the coefficients

Pointwise Multiplication Info

Properties

Pointwise multiplication:

  • Coefficients of like powers are multiplied
  • Degree can decrease
  • Commutative: P ⊙ Q = Q ⊙ P
  • Different from standard polynomial multiplication

Important: Pointwise multiplication is not the same as standard polynomial multiplication (convolution).

Input format
3x² + 4x + 5
Input: "3 4 5"
2x + 3
Input: "2 3"
x² - 1
Input: "1 0 -1"
Related functions

For standard polynomial multiplication: → Polynomial Multiplication

Formulas for pointwise multiplication

General form
\[(P \odot Q)(x) = \sum_{i=0}^{\min(m,n)} p_i \cdot q_i \cdot x^i\] Pointwise multiplication of two polynomials
Coefficient-wise
\[c_i = p_i \cdot q_i\] Coefficient for the i-th power
2D example
\[(ax + b) \odot (cx + d) = (ac)x + (bd)\] Linear polynomials
Degree of result
\[\deg(P \odot Q) \leq \min(\deg(P), \deg(Q))\] Degree can decrease

Detailed example

Example: (3x² + 4x + 5) ⊙ (2x + 3)
Given polynomials:
\[P(x) = 3x^2 + 4x + 5\] \[Q(x) = 2x + 3\]
Coefficient arrays:
P: [3, 4, 5]
Q: [2, 3, 0]
Step-by-step calculation:
Power P(x) coefficient Q(x) coefficient Product
x² 3 0 3 × 0 = 0
x¹ 4 2 4 × 2 = 8
x⁰ 5 3 5 × 3 = 15
Result:
\[(P \odot Q)(x) = 0x^2 + 8x + 15 = 8x + 15\]

Note: The x² term disappears because one of the coefficients is 0!

Comparison: Pointwise vs. Standard Multiplication

Pointwise multiplication (⊙)
\[(3x^2 + 4x + 5) \odot (2x + 3)\] \[= 0x^2 + 8x + 15\] \[= 8x + 15\]
  • Multiply coefficients of like powers
  • Degree may decrease
  • Simpler computation
Standard multiplication (×)
\[(3x^2 + 4x + 5) \times (2x + 3)\] \[= 6x^3 + 17x^2 + 22x + 15\]
  • Apply distributive law
  • Degrees add: deg(P) + deg(Q)
  • More complex computation

Applications of pointwise multiplication

Signal processing

Pointwise multiplication of polynomials corresponds to multiplying signals in the time domain.

Numerical analysis

Used in interpolation and function approximation.

Algebra

Basis for the study of polynomial rings and algebraic structures.

Mathematical properties

Basic properties
  • Commutativity: P ⊙ Q = Q ⊙ P
  • Associativity: (P ⊙ Q) ⊙ R = P ⊙ (Q ⊙ R)
  • Identity element: P ⊙ (1,1,1,...) = P
  • Zero element: P ⊙ 0 = 0
Degree properties
  • deg(P ⊙ Q) ≤ min(deg(P), deg(Q))
  • Degree can reduce due to zero coefficients
  • Usually smaller than standard multiplication
  • Zero polynomial possible for unequal degrees

Polynomial Functions

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