How to use this calculator
- Enter or choose dividend P(x).
- Enter or choose divisor D(x).
- The p(x) ÷ D(x) updates as you type. Use Copy result or Share to keep it.
Method
P(x) = D(x)·Q(x) + R(x), with deg R < deg D; each step: next quotient term = (leading term of remainder) ÷ (leading term of D)
- P
- the dividend
- D
- the divisor
- Q
- the quotient
- R
- the remainder
Worked example
- x³ ÷ x = x²; subtract x²(x − 3) = x³ − 3x² to leave x² − 4.
- x² ÷ x = x; subtract x(x − 3) = x² − 3x to leave 3x − 4.
- 3x ÷ x = 3; subtract 3(x − 3) = 3x − 9 to leave 5, whose degree is below the divisor’s.
(x³ − 2x² − 4) ÷ (x − 3) = x² + x + 3 with remainder 5.
What the result means
A zero remainder means the divisor is a factor of the dividend. The remainder always has a lower degree than the divisor.
Assumptions and limits
- Coefficients are rational numbers (decimals and fractions are kept exact).
- One variable (x) only; degree up to 50.
Questions
When should I use synthetic division instead?
Synthetic division is a shortcut that only works for divisors of the form x − r. Long division works for any divisor, such as x² + 1 or 2x + 3.
Do I need to include missing powers?
Not when you type an expression; missing powers are treated as zero. When you list coefficients, include a 0 for each missing power.