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Polynomial long division and synthetic division are algorithms for dividing one polynomial by another, producing a quotient and remainder. This is the polynomial analogue of integer long division. The result can be written as f(x) = d(x) * q(x) + r(x). This skill is essential for finding oblique asymptotes of rational functions, factoring polynomials, and performing partial fraction decomposition.
Practice long division first with clear, step-by-step layout. Introduce synthetic division as a shortcut when dividing by linear factors (x - c). Connect to the Remainder Theorem: f(c) equals the remainder when dividing f(x) by (x - c).
You already know how to manipulate variable expressions and combine like terms. Polynomial long division extends this to dividing polynomials, using the exact same logic as integer long division — the algorithm you learned for dividing numbers like 748 ÷ 23. The key insight is that both work by repeatedly asking: "how many times does the divisor fit into the leading term of what remains?"
Here is the pattern: to divide f(x) = 2x³ − 3x + 1 by d(x) = x − 2, ask how many times x goes into 2x³. The answer is 2x², so write 2x² as the first term of the quotient, multiply back (2x² · (x − 2) = 2x³ − 4x²), and subtract from f(x). This leaves 4x² − 3x + 1. Repeat: x goes into 4x² exactly 4x times, multiply back, subtract. Continue until the remainder has degree less than the divisor. The result is f(x) = (x − 2)(2x² + 4x + 5) + 11, meaning the quotient is 2x² + 4x + 5 with remainder 11. The structure is always f(x) = d(x) · q(x) + r(x) — exactly the division algorithm, a polynomial analogue of the integer fact that 748 = 23 × 32 + 12.
Synthetic division is a streamlined shortcut for the special case where the divisor is linear — specifically x − c. Instead of carrying the variable symbols, you just work with the coefficients in a row. For dividing by x − 2, you write c = 2 in the corner, list the coefficients of f(x) across the top, and proceed by bring-down, multiply-by-c, add. The last number in the row is the remainder. Synthetic division is faster once you're comfortable with it, but it only works when the divisor has the form x − c (degree 1, leading coefficient 1). Notice the sign convention: dividing by x − 2 uses c = +2, and dividing by x + 3 uses c = −3.
The Remainder Theorem connects the remainder directly to function evaluation: when f(x) is divided by x − c, the remainder equals f(c). This means synthetic division doubles as a method for evaluating polynomials at specific points. If you want to know f(7) for a degree-4 polynomial, synthetic division with c = 7 gives you the answer in fewer arithmetic operations than plugging 7 directly into the formula. This connection also underlies the Factor Theorem: x − c is a factor of f(x) if and only if f(c) = 0, which is to say the remainder is zero. Both theorems make polynomial division far more powerful than it first appears — it's not just a mechanical algorithm, but a tool for factoring and root-finding that you'll use in rational functions and partial fraction decomposition.