Polynomial Division Calculator
Divide one polynomial by another to get the quotient and remainder. Supports decimals, ± signs, and exponents like x^3. Variable is customizable (default x).
3x^3 - 2x + 1).
x or t). Case-insensitive.
Dividend = Divisor * Quotient + Remainder, formatted by (no LaTeX).
Result
Quotient
Remainder
What is a Polynomial Division Calculator?
A Polynomial Division Calculator automates the process of dividing one polynomial by another, returning the quotient and the remainder with clear, step-by-step structure. For dividend \(P(x)\) and divisor \(D(x)\neq 0\), the fundamental identity is $$P(x)=D(x)\,Q(x)+R(x),\qquad \deg R<\deg D.$$ When the divisor is linear, \(D(x)=x-c\), synthetic division streamlines computations: the remainder equals the evaluation \(R=P(c)\) (Remainder Theorem), and \(x-c\) is a factor iff \(P(c)=0\) (Factor Theorem). Beyond exact algebra, division is essential for simplifying rational expressions, partial fractions, and checking polynomial identities.
About the Polynomial Division Calculator
The calculator supports classic long division and fast synthetic division for linear divisors. It accepts missing terms (use zero coefficients), computes \(Q(x)\) and \(R(x)\), and verifies $$P(x)\stackrel{?}{=}D(x)Q(x)+R(x).$$ For teaching and readability, formulas can be rendered responsively with MathJax, while numerical coefficient handling and evaluation (e.g., \(P(c)\)) can use math.js. The same formulas displayed here are used internally, ensuring consistency between the explanation and the computed output.
How to Use this Polynomial Division Calculator
- Enter the dividend \(P(x)\) and divisor \(D(x)\). Include zero coefficients for any missing powers (e.g., \(2x^3+0x^2-5x+6\)).
- Select Long Division (general \(D(x)\)) or Synthetic Division (when \(D(x)=x-c\)).
- Run the calculation to obtain \(Q(x)\) and \(R(x)\); the tool also confirms \(P(x)=D(x)Q(x)+R(x)\).
- Interpret results: if \(R(x)=0\), then \(D(x)\) divides \(P(x)\) exactly; otherwise, the rational form is $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}.$$
Examples (using the same formulas)
Example 1 — Synthetic (linear divisor):
Divide \(P(x)=2x^3+3x^2-5x+6\) by \(D(x)=x-2\) (\(c=2\)).
Remainder \(R=P(2)=2(8)+3(4)-5(2)+6=24\).
Quotient \(Q(x)=2x^2+7x+9\); verify
$$P(x)=(x-2)(2x^2+7x+9)+24.$$
Example 2 — Long division (quadratic divisor):
\(\displaystyle \frac{x^4-1}{x^2+1}=x^2-1\) with remainder \(0\). Thus,
$$x^4-1=(x^2+1)(x^2-1).$$
Example 3 — Factor detection via remainder:
Divide \(P(x)=x^3+4x^2+5x+2\) by \(x+1\) (\(c=-1\)). Since \(P(-1)=0\), remainder \(0\) and
$$P(x)=(x+1)(x^2+3x+2).$$
6 FAQs
Q1: What’s the difference between long and synthetic division?
Long division works for any \(D(x)\); synthetic division is a shortcut when \(D(x)=x-c\).
Q2: How do I handle missing terms?
Insert zero coefficients (e.g., \(x^4+0x^3+5x-1\)) so alignment by powers is correct.
Q3: What does a nonzero remainder mean?
\(D(x)\) does not divide \(P(x)\) exactly; \(\tfrac{P}{D}=Q+\tfrac{R}{D}\) is the mixed (polynomial + proper fraction) form.
Q4: How does the Remainder Theorem help?
For \(D(x)=x-c\), the remainder equals \(P(c)\). If \(P(c)=0\), then \(x-c\) is a factor.
Q5: Can the calculator work over complex numbers?
Yes—synthetic and long division extend to complex \(c\) and complex coefficients without changing the identities.
Q6: Why is division useful for partial fractions?
Improper rational functions are first divided: \(P/D=Q+R/D\). Then \(R/D\) is decomposed into simpler fractions.