Division Algorithm for Polynomials
When you divide two integers, say 17 by 5, you get a quotient of 3 and a remainder of 2, which we write as 17 = 5 × 3 + 2. The Division Algorithm for Polynomials extends this exact idea to polynomial expressions. Given any two polynomials f(x) (the dividend) and d(x) (the divisor) with d(x) ≠ 0, there exist unique polynomials q(x) (the quotient) and r(x) (the remainder) such that f(x) = d(x) · q(x) + r(x), where either r(x) = 0 or the degree of r(x) is strictly less than the degree of d(x). This theorem is one of the foundational results of algebra. Polynomial long division is the standard procedure for finding q(x) and r(x). It mirrors integer long division step by step: you repeatedly divide the leading term of the current dividend by the leading term of the divisor, multiply, subtract, and bring down the next term. The process terminates because each step reduces the degree of the remaining expression. Mastering this algorithm is essential for factoring polynomials, evaluating limits in calculus, and simplifying rational expressions. A particularly important special case occurs when the divisor is a linear polynomial of the form (x − a). In that situation, the Remainder Theorem tells us that r(x) is simply the constant f(a), and the closely related Factor Theorem tells us that (x − a) is a factor of f(x) if and only if f(a) = 0. Synthetic division is a streamlined version of long division optimized for linear divisors. Together, these tools give high school students powerful methods for analyzing polynomial functions.
- Set Up the Division
- Divide the Leading Terms
- Multiply and Subtract
- Repeat Until Done
- Write the Final Answer
Set Up the Division
Write the dividend f(x) and divisor d(x) in standard form (descending powers of x). If any power is missing in f(x), insert a placeholder term with coefficient 0 (e.g., write x³ + 0x² − 4x + 7 instead of x³ − 4x + 7). This prevents errors when aligning like terms during subtraction.
Divide the Leading Terms
Divide the leading term of the current dividend by the leading term of the divisor. The result is the next term of the quotient q(x). For example, if the current dividend starts with 6x³ and the divisor starts with 2x, the next quotient term is 6x³ ÷ 2x = 3x².
Multiply and Subtract
Multiply the entire divisor d(x) by the quotient term you just found, then subtract that product from the current dividend. Be careful with signs when subtracting. The result is a new polynomial whose degree is strictly less than the degree of the polynomial before this step.
Repeat Until Done
Treat the result of the subtraction as the new dividend and return to Step 2. Continue until the degree of the remaining polynomial is less than the degree of the divisor. That remaining polynomial is the remainder r(x).
Write the Final Answer
Express the result as f(x) = d(x) · q(x) + r(x), or equivalently as f(x)/d(x) = q(x) + r(x)/d(x). Always verify your answer by expanding d(x) · q(x) + r(x) and confirming it equals f(x).
The Key Equation
- f(x) — Dividend (polynomial): The polynomial being divided; has degree n.
- d(x) — Divisor (polynomial): The polynomial you are dividing by; has degree m ≤ n and d(x) ≠ 0.
- q(x) — Quotient (polynomial): The result of the division; has degree n − m.
- r(x) — Remainder (polynomial): What is left over; deg(r) < deg(d), or r(x) = 0 if d(x) divides f(x) exactly.
Worked Example
Problem. Divide f(x) = 2x³ − 3x² + x + 5 by d(x) = x − 2. Express the result in the form f(x) = d(x) · q(x) + r(x).
Answer. Quotient q(x) = 2x² + x + 3, Remainder r(x) = 11
Origins · 1 minÉtienne Bézout and how this was discovered
Bézout made systematic contributions to the theory of polynomial equations. He generalized the concept of the resultant and proved Bézout's Theorem, which states that two polynomial curves of degrees m and n intersect in at most m·n points (counting multiplicity and complex intersections). His work on polynomial division and elimination theory laid groundwork that algebraists still build upon today. His textbook Cours de mathématiques was widely used in French education and influenced the formal treatment of the Division Algorithm. Bézout wrote a highly successful series of mathematics textbooks specifically for the French naval and artillery officer corps — he believed that practical soldiers needed rigorous mathematics, not just rules of thumb.
Related Concepts
- Remainder Theorem — The Remainder Theorem is a direct consequence of the Division Algorithm: when f(x) is divided by (x − a), the remainder equals f(a).
- Factor Theorem — The Factor Theorem uses the Division Algorithm to test whether (x − a) is a factor: it is a factor if and only if the remainder r = f(a) = 0.
- Synthetic Division — Synthetic division is a shorthand version of polynomial long division specialized for linear divisors of the form (x − a), using only coefficients.
- Rational Root Theorem — Once the Division Algorithm confirms a root via the Factor Theorem, repeated division is used to fully factor the polynomial into lower-degree pieces.
- Rational Expressions — Polynomial division is used to simplify improper rational expressions into a polynomial plus a proper fraction, exactly as in integer division.