Long division of polynomials looks like the long division you did in primary school, and that is exactly the trap. The steps are almost the same, but the signs, the missing terms and the stopping rule all behave differently, and one small slip early in the working spoils every line after it. Students who understand the idea still lose marks to arithmetic they would never get wrong in a simple sum.
This guide gives the rule, a reliable method, four worked examples, and a table of the ten slips that cause most of the trouble. If the vocabulary feels shaky, begin with this introduction to polynomials and the geometrical meaning of the zeroes of a polynomial, and keep the polynomials formulas and quick-revision guide open for reference.
The Division Algorithm in One Line
If p(x) and g(x) are two polynomials with g(x) ≠ 0, then you can find polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x)
where r(x) = 0, or the degree of r(x) is less than the degree of g(x). Here p(x) is the dividend, g(x) the divisor, q(x) the quotient and r(x) the remainder. It is the same statement as “dividend = divisor × quotient + remainder” for whole numbers, with one added rule about the degree of the remainder. That rule tells you when to stop.
A Method That Works Every Time
- Write both polynomials in standard form, with the powers in descending order, and insert 0 coefficients for any missing power.
- Divide the leading term of the dividend by the leading term of the divisor. That gives the first term of the quotient.
- Multiply the whole divisor by that term, and write the product under the dividend, lining up like powers.
- Subtract, changing the sign of every term of the product. Bring down the next term.
- Repeat with the new leading term.
- Stop when the degree of what is left is less than the degree of the divisor. That is the remainder.
- Check: divisor × quotient + remainder should equal the dividend.
Worked Example One: A Linear Divisor
Divide 2x² + 3x + 1 by x + 2.
- 2x² ÷ x = 2x. Multiply: 2x(x + 2) = 2x² + 4x. Subtract: (2x² + 3x) − (2x² + 4x) = −x. Bring down +1, giving −x + 1.
- −x ÷ x = −1. Multiply: −1(x + 2) = −x − 2. Subtract: (−x + 1) − (−x − 2) = 3.
- The degree of 3 is 0, which is less than the degree of x + 2, so we stop.
Quotient = 2x − 1, remainder = 3.
Check: (x + 2)(2x − 1) + 3 = 2x² − x + 4x − 2 + 3 = 2x² + 3x + 1. ✓
Worked Example Two: A Quadratic Divisor With a Missing Term
Divide x³ − 3x² + 5x − 3 by x² − 2.
The divisor has no x term, so write it as x² + 0x − 2 to keep the columns lined up.
- x³ ÷ x² = x. Multiply: x(x² − 2) = x³ − 2x. Subtract: (x³ − 3x² + 5x) − (x³ − 2x) = −3x² + 7x. Bring down −3, giving −3x² + 7x − 3.
- −3x² ÷ x² = −3. Multiply: −3(x² − 2) = −3x² + 6. Subtract: (−3x² + 7x − 3) − (−3x² + 6) = 7x − 9.
- The degree of 7x − 9 is 1, which is less than the degree of x² − 2 (which is 2), so we stop.
Quotient = x − 3, remainder = 7x − 9.
Check: (x² − 2)(x − 3) + 7x − 9 = x³ − 3x² − 2x + 6 + 7x − 9 = x³ − 3x² + 5x − 3. ✓
Worked Example Three: Finding All the Zeroes
Given that √2 is a zero of 2x⁴ − 3x³ − 3x² + 6x − 2, find the other zeroes.
Because √2 is a zero, so is −√2 for this polynomial, since it has rational coefficients, so (x − √2)(x + √2) = x² − 2 is a factor. Divide by it.
- 2x⁴ ÷ x² = 2x². Multiply: 2x²(x² − 2) = 2x⁴ − 4x². Subtract: −3x³ + x² + 6x, then bring down −2.
- −3x³ ÷ x² = −3x. Multiply: −3x(x² − 2) = −3x³ + 6x. Subtract: x² − 2.
- x² ÷ x² = 1. Multiply: x² − 2. Subtract: 0.
The quotient is 2x² − 3x + 1, which factorises as (2x − 1)(x − 1). The remaining zeroes are 1/2 and 1, together with ±√2. For the relationship between zeroes and the graph, return to the geometrical meaning of zeroes.
Worked Example Four: Working Backwards
When x³ − 3x² + x + 2 is divided by g(x), the quotient is x − 2 and the remainder is −2x + 4. Find g(x).
From p(x) = g(x) × q(x) + r(x), we get g(x) × q(x) = p(x) − r(x).
p(x) − r(x) = x³ − 3x² + x + 2 + 2x − 4 = x³ − 3x² + 3x − 2. Divide this by x − 2, and you get g(x) = x² − x + 1.
Check: (x − 2)(x² − x + 1) = x³ − x² + x − 2x² + 2x − 2 = x³ − 3x² + 3x − 2. ✓
Ten Slips and Their Fixes
| Slip | What it looks like | The fix |
| Not using standard form | Terms in a jumbled order | Rewrite in descending powers before starting |
| Skipping a missing term | x³ + 2 written without the x² and x terms | Insert 0x² and 0x |
| Subtracting without changing signs | (x² + 4x) − (x² + 2x) done as x² + 4x − x² + 2x | Change the sign of every term of the product |
| Dividing by the whole divisor | Dividing 2x² by x + 2 | Divide only by the leading term, x |
| Forgetting to bring down the next term | Working with an incomplete line | Bring down one term at a time |
| Stopping too early | Leaving a remainder of the same degree as the divisor | Continue until the remainder’s degree is smaller |
| Not stopping at all | Dividing a constant remainder further | Stop when the degree falls below the divisor’s |
| Mixing up quotient and remainder | Reporting the final line as the quotient | The last line left is the remainder |
| Not checking | Accepting any answer | Multiply back: divisor × quotient + remainder |
| Missing that remainder zero means a factor | Calling an exact division “remainder 0” and moving on | If r(x) = 0, then g(x) is a factor of p(x) |
When Division Is the Wrong Tool
Division is the right tool for finding a quotient and a remainder, and for finding the other zeroes when one factor is known. For simple questions, shorter tools may do better. The relationship between zeroes and coefficients gives the sum and product of the zeroes of a quadratic straight from the coefficients. And for a linear divisor like x − a, the remainder theorem says the remainder is simply p(a), which you can find by substitution, without dividing. Choosing the quicker method is a skill in itself, and the habit of choosing deliberately is the same one that helps in why a quadratic equation has two roots and in avoiding the common mistakes in the quadratic formula.
Polynomial Division in Your Syllabus
Polynomials is a chapter of the Class 10 Maths book in both the CBSE and the Karnataka SSLC textbooks. Published deleted-syllabus lists differ on how much of the division algorithm CBSE currently asks you to do, so check your own board’s latest syllabus for it. Practise with these chapter MCQs and read the chapter notes. In Class 11 and beyond, polynomial division reappears in algebra and calculus, which matters for JEE Main and KCET.
Frequently Asked Questions
Q1. What is the division algorithm for polynomials?
For polynomials p(x) and g(x) with g(x) not equal to zero, p(x) = g(x) × q(x) + r(x), where r(x) is zero or has a degree less than that of g(x).
Q2. When do I stop dividing?
Stop when the degree of the remaining polynomial is less than the degree of the divisor. That remaining polynomial is the remainder.
Q3. What if a power of x is missing in the polynomial?
Insert it with a zero coefficient, for example write x³ + 2 as x³ + 0x² + 0x + 2, so that like terms line up.
Q4. What does a remainder of zero mean?
It means that the divisor divides the dividend exactly, so the divisor is a factor of the dividend.
Q5. How can I check my division?
Multiply the divisor by the quotient, add the remainder, and check that you get back the original dividend.
Slow Down at the Subtraction
Almost every error in polynomial division happens in one place, the subtraction line. Change every sign, line up the powers, and verify at the end. Those three habits turn a risky topic into a dependable one.
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