synthetic division
Synthetic division is a stripped-down shortcut for the special case of dividing a polynomial by a simple linear factor of the form (x - c). It throws away the x's and works with just the coefficients, so the whole computation fits in a couple of compact rows.
The recipe: write the dividend's coefficients in descending order (using 0 for any missing power), put c in the corner, bring down the first coefficient, then repeatedly multiply by c and add down each column. The numbers you build up are the quotient's coefficients, and the very last number is the remainder.
Two limits to remember. First, it only applies to divisors of the form (x - c) — for a quadratic or higher divisor you must fall back to full long division. Second, mind the sign: dividing by (x - 3) uses c = 3, but dividing by (x + 3) means x - (-3), so you use c = -3.
Divide x^2 - 5x + 6 by (x - 2): bring down 1, 1×2=2 so -5+2=-3, -3×2=-6 so 6+(-6)=0. Quotient x - 3, remainder 0.
Only coefficients — fast, but limited to (x - c) divisors.
The last number, the remainder, also equals the polynomial evaluated at c — this is the Remainder Theorem. So a remainder of 0 means c is a root and (x - c) is a factor.