Polynomial Scalar Division

Calculator and formulas for dividing a polynomial by a real scalar

Scalar Division Calculator

What is calculated?

This function computes the scalar division of a polynomial by a real number. Each coefficient of the polynomial is divided by the divisor.

Input values

For 3x² + 4x + 5 enter: 3 4 5


Warning: Divisor must not be 0!
Result
P(x):
Divisor:
Quotient:
Each coefficient is divided by the divisor

Scalar Division Info

Properties

Scalar division:

  • Divide each coefficient by the divisor
  • Degree of the polynomial remains unchanged
  • Distributive over addition
  • Inverse of scalar multiplication

Important: Division by zero is not possible! The divisor must be non-zero.

Input format
6x² + 4x + 2
Input: "6 4 2"
4x - 2
Input: "4 -2"
2x³ + 6x
Input: "2 0 6 0"
Related functions

For other operations: → Scalar Multiplication | → Pointwise Division

Formulas for scalar division

General form
\[\frac{P(x)}{k} = \frac{1}{k} \cdot \sum_{i=0}^{n} p_i x^i = \sum_{i=0}^{n} \frac{p_i}{k} x^i\] Polynomial P(x) divided by scalar k ≠ 0
Coefficient-wise
\[c_i = \frac{p_i}{k}\] New coefficient for x^i
Distributive law
\[\frac{P + Q}{k} = \frac{P}{k} + \frac{Q}{k}\] Distributive over addition
Composite division
\[\frac{P}{k_1 \cdot k_2} = \frac{1}{k_1} \cdot \frac{P}{k_2}\] Division by a product

Detailed example

Example: (6x² + 4x + 2) ÷ 2
Given polynomial:
\[P(x) = 6x^2 + 4x + 2\] \[\text{Divisor: } k = 2\]
Coefficients:
P: [6, 4, 2]
k = 2
Step-by-step calculation:
Term Original coefficient Divisor New coefficient New term
x² 6 2 6 ÷ 2 = 3 3x²
x¹ 4 2 4 ÷ 2 = 2 2x
x⁰ 2 2 2 ÷ 2 = 1 1
Result:
\[\frac{6x^2 + 4x + 2}{2} = 3x^2 + 2x + 1\]

Note: The degree of the polynomial remains unchanged (degree 2)!

Decimal example

Example: (3x² + 4x + 5) ÷ 2
Given:
\[P(x) = 3x^2 + 4x + 5\] \[\text{Divisor: } k = 2\]
Calculation:
\[3 \div 2 = 1.5\] \[4 \div 2 = 2\] \[5 \div 2 = 2.5\]
Result:
\[\frac{3x^2 + 4x + 5}{2} = 1.5x^2 + 2x + 2.5\]

Note: Division may produce decimal coefficients!

Important restrictions

Division by zero
Not allowed:
\[\frac{P(x)}{0} = \text{undefined}\]

Division by 0 is mathematically undefined.

Be careful with small numbers:

Division by very small numbers (close to 0) can lead to numerical instability.

Applications of scalar division

Normalization

Divide by leading coefficient to normalize polynomials.

Scaling

Shrink function graphs or convert units.

Simplification

Simplify expressions by dividing out common factors.

Mathematical properties

Basic properties
  • Distributivity: (P + Q)/k = P/k + Q/k
  • Associativity: P/(ab) = (P/a)/b
  • Identity element: P/1 = P
  • Undefined: P/0 (division by zero)
Degree properties
  • deg(P/k) = deg(P) for k ≠ 0
  • Leading coefficient is divided by k
  • Zeros remain unchanged
  • Inverse of scalar multiplication

Geometric interpretation

Compression in y-direction

Division by a scalar k > 1 compresses the function graph in y-direction by factor 1/k.

\[f(x) = 4x^2 \rightarrow g(x) = \frac{4x^2}{2} = 2x^2\] Graph is compressed by factor 1/2
Reflection for negative divisor

Division by a negative scalar also reflects the graph across the x-axis.

\[f(x) = 4x^2 \rightarrow g(x) = \frac{4x^2}{-2} = -2x^2\] Graph is compressed and reflected
Special cases
k = 1:
Identity (no change)
k = -1:
Reflection across x-axis
k > 1:
Compression in y-direction
0 < k < 1:
Stretch in y-direction

Polynomial Functions

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