Polynomial Long Division
Polynomial Long Division
A method used to divide polynomials. Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper rational expression as the sum of a polynomial and a proper rational expression.
See also
Key Formula
d(x)f(x)=q(x)+d(x)r(x)
Where:
- f(x) = The dividend — the polynomial being divided
- d(x) = The divisor — the polynomial you are dividing by
- q(x) = The quotient — the result of the division
- r(x) = The remainder — what is left over, with degree less than the degree of d(x)
Worked Example
Problem: Divide (2x³ + 3x² − 5x + 6) by (x − 1).
Step 1: Divide the leading term of the dividend by the leading term of the divisor: 2x³ ÷ x = 2x². Write 2x² as the first term of the quotient.
2x3÷x=2x2
Step 2: Multiply the entire divisor by 2x² and subtract the result from the dividend.
2x2(x−1)=2x3−2x2⇒(2x3+3x2−5x+6)−(2x3−2x2)=5x2−5x+6
Step 3: Divide the new leading term by x: 5x² ÷ x = 5x. Multiply the divisor by 5x and subtract.
5x(x−1)=5x2−5x⇒(5x2−5x+6)−(5x2−5x)=6
Step 4: Divide the new leading term by x: 6 ÷ x = 6/x. Since 6 has degree 0, which is less than the degree of the divisor (degree 1), the process stops. The remainder is 6.
q(x)=2x2+5x,r(x)=6
Step 5: Write the final result using the division formula.
x−12x3+3x2−5x+6=2x2+5x+x−16
Answer: The quotient is 2x² + 5x with a remainder of 6, so (2x³ + 3x² − 5x + 6) ÷ (x − 1) = 2x² + 5x + 6/(x − 1).
Another Example
This example divides by a quadratic (not linear) divisor and requires inserting placeholder terms with coefficient 0 for missing powers of x — a crucial step students often forget.
Problem: Divide (x⁴ − 3x² + 2) by (x² + 1). Note: the dividend is missing the x³ and x terms.
Step 1: First, write the dividend with all powers of x, inserting 0 coefficients for missing terms: x⁴ + 0x³ − 3x² + 0x + 2. Divide the leading term x⁴ by x² to get x².
x4÷x2=x2
Step 2: Multiply the divisor by x² and subtract from the dividend.
x2(x2+1)=x4+x2⇒(x4+0x3−3x2+0x+2)−(x4+x2)=−4x2+2
Step 3: Divide the new leading term −4x² by x² to get −4. Multiply the divisor by −4 and subtract.
−4(x2+1)=−4x2−4⇒(−4x2+2)−(−4x2−4)=6
Step 4: The remainder 6 has degree 0, which is less than the degree of the divisor (degree 2), so the process stops.
x2+1x4−3x2+2=x2−4+x2+16
Answer: The quotient is x² − 4 with a remainder of 6, so (x⁴ − 3x² + 2) ÷ (x² + 1) = x² − 4 + 6/(x² + 1).
Frequently Asked Questions
When do you use polynomial long division?
You use polynomial long division whenever you need to divide one polynomial by another, especially when the divisor has degree 2 or higher (where synthetic division does not directly apply). Common situations include simplifying improper rational expressions, finding oblique (slant) asymptotes of rational functions, and factoring polynomials.
What is the difference between polynomial long division and synthetic division?
Synthetic division is a shortcut that works only when you divide by a linear divisor of the form (x − c). Polynomial long division works for any polynomial divisor of any degree. Both methods give the same quotient and remainder when dividing by a linear factor, but long division is more general.
How do you know when to stop dividing in polynomial long division?
You stop when the degree of the current remainder is less than the degree of the divisor. At that point, you cannot divide further, and whatever is left becomes the remainder term r(x) in the result.
Polynomial Long Division vs. Synthetic Division
| Polynomial Long Division | Synthetic Division | |
|---|---|---|
| Divisor type | Any polynomial divisor of any degree | Only linear divisors of the form (x − c) |
| Setup | Uses a bracket layout similar to numerical long division | Uses only the coefficients in a compact row-and-column format |
| Speed | More steps; slower for linear divisors | Faster and more compact for linear divisors |
| Generality | Works in all cases | Limited to dividing by (x − c) |
| Best used when | Divisor is quadratic or higher degree | Divisor is linear and you want efficiency |
Why It Matters
Polynomial long division appears throughout algebra and calculus. In precalculus, you use it to find slant (oblique) asymptotes of rational functions by dividing the numerator by the denominator. In calculus, it helps simplify integrands — you often need to rewrite an improper rational expression before applying partial fraction decomposition.
Common Mistakes
Mistake: Forgetting to include placeholder terms (with coefficient 0) for missing powers of x in the dividend or divisor.
Correction: Always write out every power of x from the highest degree down to the constant term. For example, write x³ + 2 as x³ + 0x² + 0x + 2 before dividing. Skipping a term causes columns to misalign, leading to wrong answers.
Mistake: Subtracting incorrectly — forgetting to distribute the negative sign to every term in the product.
Correction: When you subtract the product of the divisor and the current quotient term, you must change the sign of every term. A common error is flipping the sign of the first term but leaving the rest unchanged. Write out the subtraction carefully or add the opposite.
Related Terms
- Polynomial — The type of expression being divided
- Synthetic Division — A shortcut for dividing by linear factors
- Improper Rational Expression — Input expression where numerator degree ≥ denominator degree
- Proper Rational Expression — The remainder fraction produced after dividing
- Remainder Theorem — Connects remainder to evaluating f(c)
- Factor Theorem — Zero remainder means divisor is a factor
- Rational Function — Function type simplified using polynomial division
- Oblique Asymptote — Found by performing polynomial long division