Arrange the powers before dividing
Write both polynomials in descending powers of x. Insert zero coefficients for missing powers so subtraction columns line up. Divide the leading term of the current remainder by the divisor’s leading term.
Multiply that quotient term by the entire divisor, subtract the product, and bring down the next term. Repeat until the remainder degree is smaller than the divisor degree.
Check the reconstruction
Multiply the quotient by the divisor and add the remainder. You should recover the original dividend coefficient by coefficient. A zero remainder means exact divisibility.
The current Polynomial calculator offers addition, subtraction and multiplication, not division. Use it to check reconstruction; carry out long division as shown here. Synthetic division is a shortcut only for suitable linear divisors.
Write a nonzero remainder explicitly
When the remainder is nonzero, the divided expression is the quotient plus remainder/divisor. For instance, (x² + 1)/(x − 1) = x + 1 + 2/(x − 1), with x ≠ 1. The remainder is not another polynomial term to append without its denominator. Reconstructing the dividend verifies both the quotient and the sign of the remainder.
LET’S WORK IT THROUGH
Divide x² + 5x + 6 by x + 2.
- x² ÷ x = x; subtract x(x + 2) to leave 3x + 6.
- 3x ÷ x = 3; subtract 3(x + 2) to leave zero.
- The quotient terms are x and 3.
THE ANSWERQuotient x + 3; remainder 0.
YOUR TURN
A small check for understanding.
What is the remainder when x² + 1 is divided by x − 1?
Reveal the answer
2. Substituting x = 1 gives 1² + 1 = 2.
Now try your own numbers.
The polynomial calculator includes input guidance and the limits of its supported calculations. Use it alongside the method above.
Open Polynomial calculator ↗A note on precision: preserve exact fractions where possible and round numerical results at the end. Found an unclear step? Prepare a correction report.