Long Division of Polynomials: Steps, Examples, Remainder
Long Division of Polynomials: Steps, Examples, Remainder
What Is Long Division of Polynomials?
Long division of polynomials divides a polynomial, called the dividend, by a polynomial of equal or smaller degree, called the divisor, to get a quotient and a remainder. When the divisor has more than one term, long division is the standard tool, where shortcuts like synthetic division no longer apply.
The relationship it builds is the division algorithm: for a dividend P(x) and divisor D(x),
P(x) = D(x)⋅Q(x)+R(x)
where Q(x) is the quotient and R(x) is the remainder, whose degree is always less than the degree of D(x). That last condition, remainder degree below divisor degree, is the signal that tells you when to stop.
Variable Glossary:
| Symbol | Meaning |
|---|---|
| P(x) | dividend (the polynomial being divided) |
| D(x) | divisor (what you divide by) |
| Q(x) | quotient (the answer on top) |
| R(x) | remainder (what's left, lower degree than D) |
How Do You Do Long Division of Polynomials Step by Step?
The method is five repeating moves. Run them until the remainder's degree falls below the divisor's.
- Arrange both polynomials in standard form, highest power to lowest. Insert a zero coefficient for any missing power.
- Divide the leading term of the dividend by the leading term of the divisor. Write the result in the quotient.
- Multiply the whole divisor by that quotient term.
- Subtract that product from the dividend.
- Bring down the next term and repeat from step 2.
Watch it on (x2+5x+6)÷(x+2)
What Do You Do with Missing Terms?
A missing power is the single most common reason a long division goes sideways. If the dividend jumps from x4 straight to x2, the x3 term is missing, and you must hold its place with a zero coefficient before dividing.
Rewrite 6x4−9x2+3x+6 as 6x4+0x3−9x2+3x+6. The 0x3 adds nothing to the value but keeps your columns aligned, the same way a zero holds the tens place in the number 304.
Dividing by a Monomial Versus a Binomial
Not every division needs the full bracket. When the divisor is a single term, you can split the work; when it has two or more terms, the long-division layout earns its place.
- Divisor is a monomial (one term, like 2x): divide each term of the dividend separately.
(4x2−6x)÷2x=2x−3. - Divisor is a binomial or larger (like x−3 or x2+1): use the full long-division algorithm, because the terms interact and cannot be split apart.
Examples of Long Division of Polynomials
Six examples, from a clean exact division to a missing-term divide with a remainder. The problem is bold; the steps are not.
Example 1
Divide (2x2+3x−2)÷(x+2).
2x2÷x=2x
2x⋅(x+2)=2x2+4x.
(2x2+3x)−(2x2+4x)=−x.
Bring down −2: now −x−2.
−x÷x=−1.
−1⋅(x+2)=−x−2.
(−x−2)−(−x−2)=0.
Quotient 2x−1, remainder 0.
Example 2
Divide (4x2−5x−21)÷(x−3).
Correct method: subtract the entire product each time.
4x2÷x=4x.
4x⋅(x−3)=4x2−12x.
(4x2−5x)−(4x2−12x)=7x.
Bring down −21: now 7x−21.
7x÷x=7.
7⋅(x−3)=7x−21.
(7x−21)−(7x−21)=0.
Quotient 4x+7, remainder 0.
Example 3
Divide (x3+0x2+0x−8)÷(x−2).
x3÷x=x2.
x2⋅(x−2)=x3−2x2.
(x3+0x2)−(x3−2x2)=2x2.
Bring down 0: now 2x2.
2x2÷x=2x.
2x⋅(x−2)=2x2−4x.
(2x2+0)−(2x2−4x)=4x.
Bring down −8: now 4x−8.
4x÷x=4.
4⋅(x−2)=4x−8.
(4x−8)−(4x−8)=0.
Quotient x2+2, remainder 0.
Key Takeaways
- Long division of polynomials repeats divide, multiply, subtract, bring down until the remainder's degree drops below the divisor's.
- Always arrange both polynomials in standard form and insert zero placeholders for missing terms.
- The result satisfies the division algorithm P(x)=D(x)Q(x)+R(x), which you can use to check your answer.