Polynomial Pointwise Division
Calculator and example for pointwise division of two polynomials
Pointwise Division Calculator
What is calculated?
This function computes the pointwise division of two polynomials. Corresponding coefficients of like powers are divided individually.
Pointwise Division Info
Division rule
Elementwise division:
- Divide corresponding coefficients
- Like powers remain
- Variable and exponent remain
- Avoid division by zero
Important: This is not algebraic polynomial division but an elementwise operation.
Quick examples
6x² ÷ 2x² = 3x²
Same power
Same power
8x ÷ 4x = 2x
Linear terms
Linear terms
10 ÷ 5 = 2
Constant terms
Constant terms
a₀/b₀, a₁/b₁, a₂/b₂
Elementwise
Elementwise
Warning
Division by zero is not defined. Ensure that no coefficient in the divisor is zero.
Formulas for pointwise division
General form
\[P(x) \oslash Q(x) = R(x)\]
Pointwise division
Elementwise operation
\[\frac{a_i x^i}{b_i x^i} = \frac{a_i}{b_i} x^i\]
Coefficient by coefficient
Complete form
\[\sum \frac{a_i}{b_i} x^i\]
Result polynomial
Condition
\[b_i \neq 0 \; \forall i\]
No zero coefficients
Step-by-step example
Example: (3x² + 4x + 5) ÷ (2x² + 3x + 4)
1Define polynomials
\[P(x) = 3x^2 + 4x + 5\]
\[Q(x) = 2x^2 + 3x + 4\]
Dividend and divisor
2Separate coefficients
3
4
5
4
5
÷
2
3
4
3
4
Consider elementwise
3Divide individually
\[x^2: \frac{3}{2} = 1.5\]
\[x^1: \frac{4}{3} \approx 1.33\]
\[x^0: \frac{5}{4} = 1.25\]
Coefficient by coefficient
4Assemble result
\[R(x) = 1.5x^2 + 1.33x + 1.25\]
Final result
Difference to algebraic division
Pointwise division
- Divide coefficients individually
- Keep same powers
- Simple elementwise operation
- Result has same degree
\[(a_0, a_1, a_2) \oslash (b_0, b_1, b_2) = \left(\frac{a_0}{b_0}, \frac{a_1}{b_1}, \frac{a_2}{b_2}\right)\]
Algebraic division
- Polynomial long division (like numbers)
- Quotient and remainder
- Complex algorithm
- Degree is reduced
\[P(x) = Q(x) \cdot S(x) + R(x)\]
With quotient S(x) and remainder R(x)
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