Polynomial Scalar Multiplication

Calculator and formulas for multiplying a polynomial by a real scalar

Scalar Multiplication Calculator

What is calculated?

This function computes the scalar multiplication of a polynomial by a real number. Each coefficient of the polynomial is multiplied by the scalar.

Input values

For 3x² + 4x + 5 enter: 3 4 5

Result
P(x):
Scalar:
Product:
Each coefficient is multiplied by the scalar

Scalar Multiplication Info

Properties

Scalar multiplication:

  • Multiply each coefficient by the scalar
  • Degree of the polynomial remains unchanged
  • Distributive over addition
  • Associative with scalars

Tip: Scalar multiplication only changes the "scale" of the polynomial, not its "shape".

Input format
3x² + 4x + 5
Input: "3 4 5"
2x - 1
Input: "2 -1"
x³ + x
Input: "1 0 1 0"
Related functions

For other operations: → Addition | → Multiplication


Formulas for scalar multiplication

General form
\[k \cdot P(x) = k \cdot \sum_{i=0}^{n} p_i x^i = \sum_{i=0}^{n} (k \cdot p_i) x^i\] Scalar k multiplied with polynomial P(x)
Coefficient-wise
\[c_i = k \cdot p_i\] New coefficient for x^i
Distributive law
\[k \cdot (P + Q) = k \cdot P + k \cdot Q\] Distributive over addition
Associative law
\[(k_1 \cdot k_2) \cdot P = k_1 \cdot (k_2 \cdot P)\] Associative with scalars

Detailed example

Example: 2 × (3x² + 4x + 5)
Given polynomial:
\[P(x) = 3x^2 + 4x + 5\] \[\text{Scalar: } k = 2\]
Coefficients:
P: [3, 4, 5]
k = 2
Step-by-step calculation:
Term Original coefficient Scalar New coefficient New term
x² 3 2 3 × 2 = 6 6x²
x¹ 4 2 4 × 2 = 8 8x
x⁰ 5 2 5 × 2 = 10 10
Result:
\[2 \cdot (3x^2 + 4x + 5) = 6x^2 + 8x + 10\]

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

Negative scalar example

Example: -3 × (x² - 2x + 1)
Given:
\[P(x) = x^2 - 2x + 1\] \[\text{Scalar: } k = -3\]
Calculation:
\[(-3) \cdot 1 = -3\] \[(-3) \cdot (-2) = 6\] \[(-3) \cdot 1 = -3\]
Result:
\[(-3) \cdot (x^2 - 2x + 1) = -3x^2 + 6x - 3\]

Important: A negative scalar reverses all signs!

Applications of scalar multiplication

Scaling functions

Enlarge or shrink function graphs in the y-direction.

Unit conversion

Convert between measurement units in physical formulas.

Factorization

Factor out common scalars to simplify expressions.

Mathematical properties

Basic properties
  • Distributivity: k(P + Q) = kP + kQ
  • Associativity: (ab)P = a(bP)
  • Identity element: 1 · P = P
  • Zero element: 0 · P = 0
Degree properties
  • deg(k · P) = deg(P) for k ≠ 0
  • deg(0 · P) = -∞ (zero polynomial)
  • Leading coefficient is multiplied by k
  • Zeros remain unchanged

Geometric interpretation

Stretch in y-direction

Multiplication by a scalar k > 1 stretches the function graph in y-direction by factor k.

\[f(x) = x^2 \rightarrow g(x) = 2x^2\] Graph is stretched by factor 2
Reflection across x-axis

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

\[f(x) = x^2 \rightarrow g(x) = -x^2\] Graph is reflected
Important special cases
k = 1:
Identity (no change)
k = -1:
Reflection across x-axis
0 < k < 1:
Compression in y-direction
k = 0:
Zero polynomial




Polynomial Functions

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