The fast way to divide a polynomial by a linear factor (x − r). Type the coefficients (highest power first), choose the divisor, and step through it.
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⚠️ When synthetic division works — and when it doesn't
It only works when you divide by a simple linear factor, (x − r): degree one, with a coefficient of 1 in front of the x.
• If the divisor has an x² in it (a quadratic, like x² + 1 or 2x² + 5), synthetic division cannot be used — fall back to long division.
• If the coefficient isn't 1 — say (2x − 5) — you must first write it as 2(x − 2.5), divide using r = 2.5, then divide the quotient by 2.
That's exactly why we drill long division: it's the universal method, and it's the one the exam wants to see. Use synthetic division as a quick check.
The remainder equals P(r) — that's the Remainder Theorem. A remainder of 0 means (x − r) is a factor.