Mathematics
Polynomial Long Division and Synthetic Division Techniques
Quick fact
Synthetic division is a shortcut that only works when the divisor is of the form (x − c), but it can be performed in about half the steps of long division and is heavily used in calculus for evaluating polynomials.
Why this is interesting
You've never forgotten how to divide numbers, but what happens when you divide something like x³ − 2x² + 3x − 6 by x − 2? You get a polynomial quotient and a remainder—but how does that even work?
Read the full explanation
Understanding Polynomial Long Division and Synthetic Division Techniques
Polynomial long division works just like dividing numbers. You ask: 'What do I multiply the leading term of the divisor by, to get the leading term of the dividend?' You multiply, subtract, and bring down the next term. For example, dividing x² + 3x + 2 by x + 1: first, x² / x = x. Multiply (x+1) by x gives x² + x. Subtract to get 2x + 2. Then 2x / x = 2. Multiply to get 2x + 2. Subtract to get 0. So quotient is x + 2 and remainder 0. This process repeats until the degree of the remainder is less than the degree of the divisor. Synthetic division is a streamlined version that only works when the divisor is linear: (x − c). You write down the coefficients of the dividend, place c to the left, and perform a simple pattern of multiply-and-add. For example, dividing x³ − 6x² + 11x − 6 by x − 2: write coefficients 1, −6, 11, −6 and c = 2. Bring down the 1. Multiply by 2 to get 2, add to −6 to get −4, multiply by 2 to get −8, add to 11 to get 3, multiply by 2 to get 6, add to −6 to get 0. The bottom row, except the last, gives the quotient coefficients: 1, −4, 3, meaning x² − 4x + 3, and last is remainder 0.
A deeper explanation
Why does synthetic division work? It is based on the Remainder Theorem: when you divide a polynomial P(x) by (x − c), the remainder is exactly P(c). This is because P(x) = (x − c)Q(x) + R, and plugging in x = c gives P(c) = R. Synthetic division exploits this by performing the same operations as long division but in a more compact tabular form. The algorithm relies on the fact that when you subtract c times the last obtained coefficient from the next coefficient, you are essentially evaluating the polynomial using Horner's method, which is just nested multiplication from the highest degree down. This method is not only faster but also reduces the chance of arithmetic errors. Knowing this connection reveals that synthetic division is not a separate trick but a direct consequence of polynomial arithmetic. It also makes clear why it fails for divisors like (2x − 1) without a slight modification: you must first divide everything by 2 to get a divisor of the form (x − c). Understanding both methods gives you flexibility: long division works for any divisor, while synthetic division is a quick tool for evaluating polynomial remainders and roots.