Divide a polynomial by a linear binomial using synthetic division, and apply the Remainder and Factor Theorems.
Synthetic division is a fast shortcut for dividing a polynomial by (x-c): write only the coefficients, bring down the first one, then repeatedly multiply by c and add down the columns. The last number produced is the remainder; every number before it is a coefficient of the quotient, one degree lower than the original.
The Remainder Theorem ties this directly to evaluation: dividing p(x) by (x-c) always leaves a remainder equal to p(c) -- so synthetic division is secretly also a fast way to evaluate a polynomial. The Factor Theorem is the special case where that remainder is 0: (x-c) is a factor of p(x) exactly when p(c)=0, which means c is a zero of the polynomial.
Mastery looks like: They perform synthetic division correctly (including placeholder zeros), and connect the remainder to both the Remainder Theorem and the Factor Theorem.
Common stumbles: Forgetting a placeholder zero for a missing power; misreading the final remainder as part of the quotient instead of as p(c).
See it visually
Interactive, spoken-aloud tools that show this idea in motion.