introduction
Prepare Class 11 Mathematics Unit 5: Polynomials with complete solved exercises from HSA Notes. These notes are prepared according to the FBISE NBF New Book and New Course for Session 2026–27 and cover Exercises 5.1, 5.2, 5.3, and the Review Exercise.
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Polynomials are an important part of Class 11 Mathematics because they develop students’ understanding of algebraic expressions, polynomial functions, division, factors, zeros, and equations. This unit also introduces important techniques such as the Remainder Theorem, Factor Theorem, synthetic division, and factorization. The final exercise applies polynomial concepts to practical problems involving dimensions, area, volume, and other situations.
HSA Notes provides easy, exam-focused solutions so students can understand the mathematical method used in each question and revise important concepts efficiently.
HSA Notes Features
- Complete Exercises 5.1–5.3 and Review Exercise
- Based on the FBISE NBF New Book
- Updated for Session 2026–27
- Short and exam-focused solutions
- Remainder Theorem and Factor Theorem
- Synthetic division
- Polynomial factorization
- Finding zeros and factors
- Algebraic problem solving
- Applications involving area and volume
- Important MCQs and conceptual questions
- Detailed mathematical working
- PDF with diagrams and solved examples
Unit 5: Polynomials
A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer powers of variables. In this unit, students learn how to work with polynomial expressions and functions and how to determine their zeros and factors.
The unit develops important algebraic techniques that are useful throughout higher mathematics. Students should understand the relationship between zeros, factors, divisibility, and remainders and practice applying the appropriate theorem to each question.
Exercise 5.1 Remainder Theorem
Exercise 5.1 introduces the Remainder Theorem and teaches students how to determine the remainder obtained when a polynomial is divided by a linear expression.
Selected solved results include:
Q1(i) Remainder = 5
Q1(ii) Remainder = 35
Students also learn to test whether a given expression is a factor of a polynomial by evaluating the polynomial at the appropriate value.
Q2 x − 3 is factor (P(3) = 0)
Q3 x − 3 not factor (P(3) = −5)
Other important results include:
Q5 q = −12
Q6 m = 6
Q7 1 is zero, −2 is not
Q8 Zeros: 2, −½, −3
Students also practice writing a polynomial in factored form.
Q9 f(x) = (x − 4)(x² + 3x − 2)
The exercise concludes with application-based polynomial problems.
Q10 Floor area = x² + 10x + 24
Understanding the Remainder Theorem is essential because it provides a quick method for evaluating remainders without carrying out complete polynomial division.
Exercise 5.2 Factor Theorem
Exercise 5.2 focuses on the Factor Theorem, which provides a useful method for determining whether a linear expression is a factor of a polynomial. Students also use the theorem to completely factor polynomial expressions.
Selected answers include:
Q1 (y + 1)(y − 3)(y + 2)
Q2 (x − 1)(2x − 1)(x + 1)
Q3 (x − 2)(2x + 3)(x + 3)
Q4 (x − 3)(3x² + 4x + 12)
Q5 (t − 1)(t² + 2t + 5)
Q6 (x − 2)(2x + 1)(x − 6)
Q7 (2x − 1)(x − 2)(x − 5)
Students should carefully practice substitution and factorization because sign errors can change the final answer. The complete notes provide the required mathematical working for these questions.
Exercise 5.3 Applications of Polynomials
Exercise 5.3 applies polynomial concepts to practical mathematical situations. These questions involve dimensions, area, volume, tickets, lengths, and sides.
Selected solved answers include:
Q1 Dimensions: 12 × 2 × 5 cm
Q2 Tickets sold in 12th game = 650
Q3 Dimensions: 8 × 6 × 3
Q4 Dimensions: 25 × 11 × 9
Q5 Length ACED = 3x + 7; Area square = (x + 4)²
Q6 Sides: y − 1, y + 1
These application questions help students understand how algebraic expressions and polynomial relationships can be used to solve real-world mathematical problems.
Review Exercise 5
The Review Exercise combines the major concepts studied throughout Unit 5. Students should revise the Remainder Theorem, Factor Theorem, polynomial division, zeros, factors, and applications before attempting the review questions.
Important review answers include:
(i) (x + 1)(x − 2)
(ii) Remainder = 0
(iii) −3
(iv) Divisor & factor
(v) k = −20
(vi) Remainder
(vii) p = 1
(viii) f(2) = 12
These questions provide useful revision before the examination and help students identify which concepts require additional practice.
Synthetic Division and Factorization
Synthetic division is an efficient method for dividing a polynomial by a linear factor. Students should learn how to organize the coefficients correctly and interpret the quotient and remainder.
Factorization is another major skill in this unit. Once a zero or factor is identified, the polynomial can often be reduced to a lower-degree expression and further factorized. HSA Notes explains these methods in an easy format for Class 11 students.
FBISE Exam Preparation
For the FBISE Class 11 Mathematics examination, students should focus on understanding the theorems rather than simply memorizing answers. Practice substitution carefully, check signs, and verify factors by evaluating the polynomial.
Important preparation areas include Remainder Theorem, Factor Theorem, zeros of polynomials, synthetic division, factorization, and application-based numerical questions.
Students should also revise the Review Exercise after completing Exercises 5.1–5.3.
View the Complete PDF
The complete Class 11 Math Unit 5 Polynomials solved PDF contains detailed solutions, synthetic division, factorization, mathematical working, diagrams, and selected key answers for Exercises 5.1, 5.2, 5.3, and the Review Exercise.
Who Can Use These Notes?
These notes are useful for students preparing for FBISE and Punjab Board examinations. They can support students studying in schools such as Beaconhouse, Roots, LGS, City School, Karachi Grammar, Aitchison, Lawrence, ICG, APS, and FPS.
The notes can be used for classroom revision, homework, independent study, numerical practice, and examination preparation.
Join Classes and Get Support
Students who need additional support with Class 11 Mathematics can use HSA Notes for exercise practice and revision. Regular practice of textbook questions can help students improve their understanding of polynomial concepts and mathematical working.
Email: hsanotes48@gmail.com
Conclusion
Class 11 Mathematics Unit 5: Polynomials develops essential algebraic skills through polynomial functions, remainders, factors, zeros, synthetic division, factorization, and practical applications. The concepts introduced in this unit also provide a foundation for more advanced mathematics.
HSA Notes provides complete solved Unit 5 Polynomials notes for the FBISE NBF New Book, Session 2026–27, covering Exercises 5.1–5.3 and the Review Exercise. Use these notes with the official textbook to revise concepts, practice calculations, understand theorem-based questions, and strengthen your preparation for the Class 11 Mathematics examination.