Dividing Polynomials by Long Division
Simple Explanation
Polynomial long division works just like ordinary long division with numbers: divide, multiply, subtract, bring down — repeated until the remainder has a lower degree than the divisor. It writes any polynomial P(x) as P(x) = D(x)·Q(x) + R(x).
Why Do We Need It?
Dividing polynomials is the foundation for factoring higher-degree polynomials, simplifying rational expressions, and — as this whole chapter builds toward — finding a polynomial's roots efficiently.
Formula
The Polynomial Division Identity
P(x) = D(x) · Q(x) + R(x), where the degree of R(x) is less than the degree of D(x)
Dividing a polynomial P(x) by a divisor D(x) always produces a quotient Q(x) and a remainder R(x), related by this identity — the same "dividend = divisor × quotient + remainder" pattern used for ordinary numbers.
- P(x)
- — the dividend — the polynomial being divided
- D(x)
- — the divisor — the polynomial being divided by
- Q(x)
- — the quotient
- R(x)
- — the remainder (degree strictly less than D(x), or R(x)=0 if it divides evenly)
When to use it: Whenever you divide one polynomial by another and need to check or express the result correctly.
Worked Example
Divide a polynomial by a linear divisor
Divide 2x³ − 3x² − 11x + 6 by (x − 3) using long division.
Why Does This Work?
At each step, subtracting a multiple of the divisor that matches the current leading term removes that term entirely, always leaving a lower-degree remainder — repeating until the remaining degree is less than the divisor's degree reconstructs exactly P(x) = D(x)Q(x) + R(x).
Real-Life Example
Simplifying a rational design formula
An engineer has a complicated polynomial formula divided by a simpler linear expression and wants to simplify it for a spreadsheet.
Long division rewrites the complicated fraction as a simpler polynomial plus a small remainder term — a genuine simplification for reuse.
Practice
Divide x² + 5x + 6 by (x + 2). What is the quotient?
MediumCommon mistake
Forgetting to bring down the next term with its correct sign after each subtraction step — a sign error at any stage propagates through every later step of the division.
Quick Review
- P(x) = D(x)·Q(x) + R(x), with degree(R) < degree(D).
- Divide, multiply, subtract, bring down — repeat until finished.
- A remainder of 0 means the divisor divides P(x) exactly.