Polynomial Subtraction

Calculator and formulas for subtraction of two polynomials

Polynomial Subtraction Calculator

What is calculated?

This function computes the subtraction of two polynomials. The corresponding coefficients of like powers are subtracted from each other.

Enter polynomials

For 3x² + 4x + 5 enter: 3 4 5


For 2x + 3 enter: 2 3
Result
P(x):
Q(x):
Difference:
Corresponding coefficients are subtracted from each other

Polynomial subtraction info

Properties

Polynomial subtraction:

  • Subtract coefficients of like powers
  • Degree equals max(deg(P), deg(Q))
  • Not commutative: P - Q ≠ Q - P
  • Inverse of addition

Important: When subtracting, the signs of the subtrahend change!

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

For other operations: → Addition | → Multiplication

Formulas for polynomial subtraction

General form
\[P(x) - Q(x) = \sum_{i=0}^{\max(m,n)} (p_i - q_i) x^i\] Subtraction of two polynomials
Coefficient-wise
\[c_i = p_i - q_i\] Difference of coefficients for x^i
Sign rule
\[P - Q = P + (-Q)\] Subtraction as addition of the negative
Non-commutativity
\[P - Q = -(Q - P)\] Order matters

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: [0, 2, 3] (with 0 for x²)
Step-by-step calculation:
Power P(x) coefficient Q(x) coefficient Difference New term
x² 3 0 3 - 0 = 3 3x²
x¹ 4 2 4 - 2 = 2 2x
x⁰ 5 3 5 - 3 = 2 2
Result:
\[(3x^2 + 4x + 5) - (2x + 3) = 3x^2 + 2x + 2\]

Note: The degree remains 2 since the x² term is preserved!

Sign change example

Example: (x² + 2x - 1) − (3x² - x + 4)
Given:
\[P(x) = x^2 + 2x - 1\] \[Q(x) = 3x^2 - x + 4\]
Expand the parentheses:
\[P - Q = P + (-Q)\] \[(-Q) = -3x^2 + x - 4\]
Coefficient-wise subtraction:
Power P(x) Q(x) P - Q
x² 1 3 1 - 3 = -2
x¹ 2 -1 2 - (-1) = 3
x⁰ -1 4 -1 - 4 = -5
Result:
\[(x^2 + 2x - 1) - (3x^2 - x + 4) = -2x^2 + 3x - 5\]

Important: Pay attention to sign changes when subtracting!

Non-commutativity of subtraction

Important: P - Q ≠ Q - P
P - Q:
\[(3x + 2) - (x + 1)\] \[= 3x + 2 - x - 1\] \[= 2x + 1\]
Q - P:
\[(x + 1) - (3x + 2)\] \[= x + 1 - 3x - 2\] \[= -2x - 1\]
Relation:
\[Q - P = -(P - Q)\] \[-2x - 1 = -(2x + 1)\]

The results are opposite to each other!

Applications of polynomial subtraction

Error analysis

Compute the difference between theoretical and measured values.

Signal processing

Remove noise by subtracting known signals.

Simplification

Simplify complex expressions by cancelling like terms.

Mathematical properties

Basic properties
  • Non-commutativity: P - Q ≠ Q - P
  • Associativity: (P - Q) - R = P - (Q + R)
  • Identity element: P - 0 = P
  • Inverse: P - P = 0
Degree properties
  • deg(P - Q) ≤ max(deg(P), deg(Q))
  • Degree can decrease due to cancellation
  • Leading coefficients are subtracted
  • Inverse of addition

Geometric interpretation

Function difference

The subtraction P(x) - Q(x) creates a new function representing the vertical difference between P and Q.

\[f(x) = (x^2) - (x) = x^2 - x\] Parabola minus line
Zeros

The zeros of P - Q correspond to intersection points of the graphs of P and Q.

\[P(x) - Q(x) = 0\] \[\Rightarrow P(x) = Q(x)\] Intersection points of the functions
Practical significance

In physics and engineering, polynomial subtraction is often used to compute deviations, errors or differences between models. For example, the difference between a theoretical and experimental polynomial can reveal systematic errors.

Polynomial Functions

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