Introduction
The best Unit 9, Division of Polynomials, solved notes for Class 11 Punjab Boards are now available, with 100 percent of exercises solved using easy, step by step solutions and the marking scheme, based on the new book from PECTAA, the Punjab Curriculum and Textbook Board. These notes are perfect for the 2026 exams and are built to help students score 90 plus marks.
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What’s Inside Unit 9, Dividing Polynomials the Fast and the Long Way
Unit 9 takes a single operation, dividing one polynomial by another, and shows there’s more than one road to the same answer, with each route suited to a different kind of problem.
It opens with long division of polynomials, the direct method of dividing a polynomial by another polynomial to find a quotient and a remainder, following the same logic as long division with numbers but applied to terms with variables and exponents.
From there, the unit introduces synthetic division, a faster shorthand method for dividing a polynomial by a linear factor of the form (x − a), using only the coefficients of the polynomial to arrive at the same quotient and remainder with far less writing.
This sets up the Remainder Theorem, which states that the remainder when a polynomial P(x) is divided by (x − a) is simply P(a), letting students find a remainder by direct substitution instead of carrying out the full division.
The unit closes with the Factor Theorem, the natural extension of the Remainder Theorem: if P(a) = 0, then (x − a) is a factor of P(x). This is used to test possible factors, factorize higher degree polynomials, and solve polynomial equations that don’t factor by inspection.
Exercise 9.1, Remainder and Quotient by Division, Fully Solved
All questions in Exercise 9.1 are fully solved, covering long division of a polynomial by both linear and quadratic divisors to find the quotient and remainder.
| Question | Division | Result |
|---|---|---|
| Q1(i) | (3x² − x + 2) ÷ (x − 1) | Quotient 3x + 2, Remainder 4 |
| Q1(ii) | (x³ + 12x² − 3x + 4) ÷ (x − 2) | Quotient x² + 14x + 25, Remainder 54 |
| Q1(iii) | (x⁴ − 5x³ − 8x² + 13x + 12) ÷ (x − 6) | Quotient x³ + x² − 2x + 1, Remainder 18 |
| Q1(iv) | (5x⁴ − 3x³ + 2x² − 1) ÷ (x² + 4) | Quotient 5x² − 3x − 18, Remainder 12x + 71 |
| Q1(v) | (3x⁴ − 5x³ + 4x − 6) ÷ (x² − 3x + 5) | Quotient 3x² + 4x − 7, Remainder 26x − 41 |
The full solved PDF for Exercise 9.1 is available, covering every question and part in complete detail.
Exercise 9.2, Synthetic Division, the Remainder Theorem, and the Factor Theorem, Fully Solved
Exercise 9.2 is fully solved, covering synthetic division by a linear factor, finding remainders directly using the Remainder Theorem, applying the Factor Theorem to test and confirm factors, and using both theorems together to factorize polynomials and solve polynomial equations.
Review Exercise, Unit 9, Fully Solved
The Review Exercise for Unit 9 is fully solved, bringing together long division, synthetic division, and both theorems in mixed, exam style questions that mirror how the topic is actually tested.
Bonus Resources
- A Division of Polynomials formula sheet PDF for quick reference.
- The top 30 long division and synthetic division shortcuts for faster problem solving.
- Division of Polynomials video lectures.
View the Full Solved PDF
The complete solved PDF for Unit 9 is available to view for free.
Why This Unit Matters for Your Overall Score
Division of Polynomials is a unit where speed and accuracy come from knowing which method fits the question in front of you, rather than defaulting to long division every time. Exercise 9.1 builds the foundation by making sure long division itself is solid, including the trickier cases where the divisor is quadratic rather than linear. Once that’s comfortable.
Exercise 9.2 shows how much time synthetic division and the Remainder Theorem save when the divisor is linear, which is the far more common case on exam papers. The Factor Theorem then turns this into a factorization tool, and that skill carries forward directly into later units on polynomial equations and curve sketching, making Unit 9 one of the more quietly high leverage units in the syllabus.
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Frequently Asked Questions
Which exercises are covered in these Unit 9 notes?
Exercise 9.1 and Exercise 9.2 are covered in full, along with the Unit 9 Review Exercise, spanning long division, synthetic division, the Remainder Theorem, and the Factor Theorem.
What’s the difference between the Remainder Theorem and the Factor Theorem?
The Remainder Theorem tells you the remainder of P(x) ÷ (x − a) without dividing, by evaluating P(a). The Factor Theorem is the special case where that remainder is zero: if P(a) = 0, then (x − a) is a factor of P(x), which means the division comes out exact.
Are past papers included with these notes?
Yes, Unit 9 past papers solved from 2018 through 2025 are included as a bonus resource.
Do I need to download the PDF to view these notes?
No, the full solved PDF can be viewed directly on the site for free, with no download required.
Conclusion
With Exercise 9.1, Exercise 9.2, and the Review Exercise fully solved, Class 11 students across all nine Punjab boards have a complete, exam ready resource for mastering Division of Polynomials ahead of the 2026 exams. Use these notes alongside the bonus formula sheet, shortcuts, and past papers to build both speed and accuracy for your Mathematics exam.
Best of luck for your Class 11 Mathematics exam!