Long division, polynomial style
Polynomial long division follows the very rhythm you learned for numbers: divide, multiply, subtract, bring down — repeat. Each round, you divide the leading term of what remains by the leading term of the divisor, write that on top, multiply it back through, and subtract to shrink the remainder.
- Put both polynomials in standard form; fill any missing power with a 0 placeholder.
- Divide: leading term of the dividend ÷ leading term of the divisor → next quotient term.
- Multiply that quotient term by the whole divisor and subtract it from the current dividend.
- Bring down the next term and repeat until the remainder’s degree is below the divisor’s.
(x^2 + 5x + 6) ÷ (x + 2)
x + 3
__________________
x + 2 ) x^2 + 5x + 6
x^2 + 2x ← (x)(x+2)
--------
3x + 6
3x + 6 ← (3)(x+2)
------
0 remainder
Result: x + 3 (remainder 0, so x+2 divides evenly)Remainders and the division algorithm
Division does not always come out clean. When a remainder is left, we record the result exactly as with whole numbers — the division algorithm in polynomial form: dividend = divisor × quotient + remainder. A remainder of 0 is special: it means the divisor is a factor, and the matching value is a root of the polynomial.
(2x^2 + 3x − 4) ÷ (x + 3)
2x − 3
__________________
x + 3 ) 2x^2 + 3x − 4
2x^2 + 6x
--------
−3x − 4
−3x − 9
-------
5 remainder
Result: 2x − 3 + 5/(x + 3)
Check: (x+3)(2x−3) + 5 = 2x^2 + 3x − 9 + 5 = 2x^2 + 3x − 4 ✓Synthetic division: the shortcut
When the divisor has the form x − c, synthetic division does the same job with just the coefficients — no x’s to carry. Write the dividend’s coefficients in standard form (zeros for gaps), put c to the left, then bring down, multiply, add, repeat. The last number is the remainder; the rest are the quotient’s coefficients, one degree lower.
Divide x^3 − 4x^2 + 0x + 6 by x − 3 (so c = 3)
3 | 1 −4 0 6
| 3 −3 −9
+---------------------
1 −1 −3 −3
\---quotient---/ \rem/
Quotient: x^2 − x − 3, remainder −3
So x^3 − 4x^2 + 6 = (x − 3)(x^2 − x − 3) − 3