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I. Chapter Summary

The chapter Polynomials introduces students to algebraic expressions involving variables and exponents. It focuses on understanding what polynomials are, their types, the degree of a polynomial, and how to add, subtract, and multiply them. The chapter also explains the Remainder Theorem, Factor Theorem, and how to find zeroes of a polynomial. Students explore standard identities and apply them to simplify and solve algebraic expressions. This chapter builds the foundation for advanced algebra in higher classes.

II. Key Concepts Covered

Concept	Explanation
Algebraic Expression	Expression made up of variables, constants, and operations.
Polynomial	An expression of the form $a_n x^n + a_{n-1} x^{n-1} + dots + a_0$, where powers of x are whole numbers.
Degree of a Polynomial	Highest power of the variable.
Types of Polynomials
→ Monomial: one term (e.g., 3x)
→ Binomial: two terms (e.g., x + 5)
→ Trinomial: three terms (e.g., $x^2 + x + 1$)
→ Zero Polynomial: constant 0
→ Linear, Quadratic, Cubic: degree 1, 2, 3
Addition/Subtraction/Multiplication	Performed by combining like terms or using distributive property.
Remainder Theorem	If $p(x)$ is divided by $(x – a)$, remainder $= p(a)$.
Factor Theorem	If $text{If } p(a) = 0, text{ then } (x – a) text{ is a factor of } p(x)$ .
Zeroes of a Polynomial	Value(s) of variable for which $p(x) = 0$.
Algebraic Identities
$(x + y)^2 = x^2 + 2xy + y^2$
$(x – y)^2 = x^2 – 2xy + y^2$
$x^2 – y^2 = (x + y)(x – y)$
$(x + a)(x + b) = x^2 + (a + b)x + ab$

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. What is the degree of the polynomial $3x^2 + 5x – 4$?
   a) 0
   b) 1
   c) 2 ✔️
   d) 3
2. Which of the following is a binomial?
   a) $3x$
   b) $x + 1$ ✔️
   c) $x^2 + x + 1$
   d) 0
3. If $p(x) = x^2 – 4x + 3$, what is p(1)?
   a) 0
   b) 1 ✔️
   c) $–1$
   d) 2
4. $(x + y)^2 – (x – y)^2$?
   a) $4xy$ ✔️
   b) $2x^2$
   c) $x^2 – y^2$
   d) $x^2 + y^2$
   (PYQ 2020)

(B) Short Answer Questions (2/3 Marks)

1. Classify the polynomial $x^3 – 4x + 6$ by the number of terms and degree.
2. Find the value of p(2), if $p(x) = x^2 – 3x + 2$. (PYQ 2019)
3. Divide $frac{2x^2 + 3x – 5}{x + 2}$ using long division.
4. Verify $(x + y)^2 = x^2 + 2xy + y^2$ for $x = 3$ and $y = –1$.

(C) Long Answer Questions (5 Marks)

1. Using Remainder Theorem, find the remainder when $p(x) = x^3 – 2x^2 + x – 1$  is divided by $x – 1$. Also verify using division. (PYQ 2020)
2. Show that $(x + 2)(x – 2) = x^2 – 4$ and use it to evaluate (103 × 97).
3. Factorize $x^2 + 7x + 12$ using the factor theorem.
4. Multiply $(x + 3)(x – 2)(x + 1)$ and simplify the expression.

(D) HOTS (Higher Order Thinking Skills)

1. If $p(x) = x^3 + ax^2 + bx + c$ has 1 as a root and leaves remainder 3 when divided by $x – 2$, find c.
2. Find a polynomial of degree 2 whose zeroes are 2 and –3.

IV. Key Formulas/Concepts

Concept	Formula / Identity
$(x + y)^2$	$= x^2 + 2xy + y^2$
$(x – y)^2$	$= x^2 – 2xy + y^2$
$x^2 – y^2$	$= (x + y)(x – y)$
$(x + a)(x + b)$	$= x^2 + (a + b)x + ab$
Remainder Theorem	If $p(x)$ is divided by $x – a$, remainder $= p(a)$
Factor Theorem	If $p(a) = 0$, then $(x – a)$ is a factor

V. Deleted Portions (CBSE 2025–2026)

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Chapter	Estimated Marks	Type of Questions Typically Asked
Polynomials	6–8 Marks	1 MCQ, 1 Short Answer, 1 Long Answer, 1 HOTS or Identity-Based Question

VII. Previous Year Questions (PYQs)

Marks	Question	Year
1 mark	Expand $(x + y)^2$ using identity.	PYQ 2020
2 marks	Evaluate $p(-1)$ if $p(x) = x^2 + 2x + 1$.	PYQ 2018
3 marks	Classify and find degree of $3x^5 – 7x^2 + 4$.	PYQ 2019
5 marks	Use Remainder Theorem to find remainder and verify.	PYQ 2020

VIII. Real-World Application Examples

• Engineering Design: Polynomial curves are used in designing bridges and roads.
• Computer Graphics: Polynomial equations help in curve modeling and animation paths.
• Economics/Finance: Revenue and cost functions often follow polynomial models.
• Physics: Projectile motion and acceleration graphs are modeled by quadratic polynomials.

IX. Student Tips & Strategies for Success

Time Management

• Allocate 3–4 days:
  Day 1: Basics and classification
  Day 2: Operations and identities
  Day 3: Factor Theorem and remainder
  Day 4: Practice and PYQs

Exam Preparation

• Use tables to compare identities and their applications.
• Practice division and verification techniques for Remainder Theorem.

Stress Management

• Use color-coded formulas, sticky notes, and flashcards.
• Try peer teaching — explain a concept to a friend for better recall.

X. Career Guidance & Exploration

• For Classes 9–10:
  ➤ Start building algebraic reasoning early.
  ➤ Attempt NTSE, Olympiads (SOF IMO), and RMO.
• For Classes 11–12:
  ➤ Algebra is critical in JEE, CUET, NEET (for Physics)
  ➤ Careers in:
  • Engineering
  • Finance and Actuarial Science
  • Data Science
  • Machine Learning

XI. Important Notes

 Practice all standard identities and use them to simplify expressions.
 Don’t just memorize theorems—understand the logic behind them.
 For accuracy, always verify answers using substitution.
 Refer to CBSE Sample Papers and NCERT Exemplar for higher-order problem practice.