CBSE Class 9 Mathematics Polynomials: Complete Notes and Practice Problems (2025–2026 Guide)

Mastering CBSE Class 9 Mathematics polynomials is a fundamental milestone for scoring top marks in your 2025–2026 academic assessments and building an unshakeable foundation for Class 10 algebra. Chapter 2 of the NCERT syllabus transitions students from basic arithmetic expressions to structured algebraic thinking. This comprehensive guide covers all core definitions, classifications, theorems, algebraic identities, real-world applications, and solved exam-standard problems tailored for CBSE Class 9 students.

Key Concepts in CBSE Class 9 Polynomials

A solid grasp of polynomial theory requires understanding how algebraic expressions are structured, evaluated, and factored. Below is the detailed breakdown of the complete CBSE syllabus.

1. What is a Polynomial in One Variable?

An algebraic expression of the form:

p(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₁x + a₀

is called a polynomial in one variable x, where:

• a_n, a_n−1, ..., a₀ are real numbers (called coefficients).
• a_n ≠ 0 (a_n is the leading coefficient).
• n is a whole number (non-negative integer: 0, 1, 2, 3, ...).

Crucial Rule for CBSE Exams: An expression is NOT a polynomial if any variable has a negative exponent, fractional exponent, or appears in the denominator or under a radical. For example:

• 2x² + 5x − 3 is a polynomial (exponents 2, 1, 0 are whole numbers).
• √x + 4 = x^1/2 + 4 is not a polynomial (exponent 1/2 is not a whole number).
• 3/x + 2 = 3x⁻¹ + 2 is not a polynomial (exponent −1 is negative).

2. Degree of a Polynomial

The degree of a non-zero polynomial is the highest power of the variable present in the expression.

• Constant Polynomial: Degree is 0 (e.g., p(x) = 7 = 7x⁰).
• Zero Polynomial: The constant polynomial 0 has an undefined degree.
• Linear Polynomial: Degree is 1 (e.g., p(x) = 3x − 5).
• Quadratic Polynomial: Degree is 2 (e.g., p(x) = 4x² + 7x − 2).
• Cubic Polynomial: Degree is 3 (e.g., p(x) = x³ − 2x² + 3x − 1).

3. Classification of Polynomials

Polynomials are categorized based on both the number of terms and their degree:

4. Zeroes of a Polynomial

A real number k is called a zero (or root) of a polynomial p(x) if and only if p(k) = 0.

• A non-zero constant polynomial has no zero.
• Every linear polynomial in one variable has a unique zero (for ax + b = 0, zero is x = −b/a).
• A polynomial of degree n can have at most n real zeroes.

5. The Remainder Theorem and Factor Theorem

The Remainder Theorem: Let p(x) be any polynomial of degree greater than or equal to 1, and let a be any real number. If p(x) is divided by the linear polynomial (x − a), then the remainder is p(a).

The Factor Theorem: If p(x) is a polynomial of degree n ≥ 1 and a is any real number:

1. (x − a) is a factor of p(x) if p(a) = 0.
2. Conversely, p(a) = 0 if (x − a) is a factor of p(x).

6. Complete List of Standard NCERT Algebraic Identities

You must memorize these 8 fundamental algebraic identities for simplification and factorisation:

1. Identity I: (x + y)² = x² + 2xy + y²
2. Identity II: (x − y)² = x² − 2xy + y²
3. Identity III: x² − y² = (x + y)(x − y)
4. Identity IV: (x + a)(x + b) = x² + (a + b)x + ab
5. Identity V: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
6. Identity VI: (x + y)³ = x³ + y³ + 3xy(x + y) = x³ + 3x²y + 3xy² + y³
7. Identity VII: (x − y)³ = x³ − y³ − 3xy(x − y) = x³ − 3x²y + 3xy² − y³
8. Identity VIII: x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx)

Special Conditional Property: If x + y + z = 0, then x³ + y³ + z³ = 3xyz. This property is frequently tested in CBSE annual school examinations!

7. Real-World Applications of Polynomials

Polynomials are not merely abstract formulas; they govern critical computational models across disciplines:

• Kinematics & Physics: The trajectory of a launched cricket ball follows a quadratic height model: h(t) = −½gt² + v₀t + h₀.
• Financial Forecasting: Business profit functions are calculated using revenue and cost polynomials: P(x) = R(x) − C(x).
• Engineering & Design: Structural deflection curves for bridges and roller coasters use cubic spline polynomials for smooth load distribution.

Important CBSE Questions with Step-by-Step Solutions

Here are high-yield questions structured strictly according to the latest CBSE Mathematics pattern for Class 9.

Question 1 (1 Mark - Concept Check)

Question: Find the value of the polynomial p(x) = 5x − 4x² + 3 at x = −1.

Solution:

Given: p(x) = −4x² + 5x + 3

Substitute x = −1 into the polynomial:

p(−1) = 5(−1) − 4(−1)² + 3
p(−1) = −5 − 4(1) + 3
p(−1) = −5 − 4 + 3
p(−1) = −9 + 3 = −6

Answer: The value of the polynomial at x = −1 is −6.

Question 2 (2 Marks - Factor Theorem Application)

Question: Find the value of k if (x − 1) is a factor of p(x) = 2x² + kx + √2.

Solution:

1. By the Factor Theorem, if (x − 1) is a factor of p(x), then p(1) = 0.
2. Evaluate p(1):
   p(1) = 2(1)² + k(1) + √2 = 0
   2(1) + k + √2 = 0
   2 + k + √2 = 0
3. Isolate k:
   k = −(2 + √2) = −2 − √2

Answer: The value of k is −(2 + √2).

Question 3 (3 Marks - Factorisation by Splitting the Middle Term)

Question: Factorise the quadratic polynomial: 6x² + 17x + 5.

Solution:

1. Compare with standard form ax² + bx + c: here a = 6, b = 17, c = 5.
2. Find two numbers p and q such that:
   p + q = b = 17
   p × q = a × c = 6 × 5 = 30
3. The factors of 30 that add up to 17 are 15 and 2 (since 15 × 2 = 30 and 15 + 2 = 17).
4. Split the middle term:
   6x² + 17x + 5 = 6x² + 15x + 2x + 5
5. Group terms into pairs and factor out common terms:
   = 3x(2x + 5) + 1(2x + 5)
6. Factor out the common binomial (2x + 5):
   = (2x + 5)(3x + 1)

Answer: 6x² + 17x + 5 = (2x + 5)(3x + 1).

Question 4 (4 Marks - Cubic Factorisation)

Question: Factorise the cubic polynomial: p(x) = x³ − 23x² + 142x − 120.

Solution:

1. Step 1: Find one root by trial (Rational Root Theorem)
   Factors of the constant term (−120) include ±1, ±2, ±3, ±4, ±5, ±6...
   Let us test x = 1:
   p(1) = (1)³ − 23(1)² + 142(1) − 120
   p(1) = 1 − 23 + 142 − 120 = 143 − 143 = 0
   Since p(1) = 0, by Factor Theorem, (x − 1) is a factor of p(x).
2. Step 2: Divide p(x) by (x − 1) using polynomial long division
   Dividing x³ − 23x² + 142x − 120 by (x − 1) gives:
   Quotient = x² − 22x + 120
3. Step 3: Factorise the quadratic quotient
   q(x) = x² − 22x + 120
   Find two numbers whose product is 120 and sum is −22: these are −12 and −10.
   x² − 12x − 10x + 120 = x(x − 12) − 10(x − 12) = (x − 12)(x − 10)
4. Step 4: Combine all linear factors
   p(x) = (x − 1)(x − 10)(x − 12)

Answer: x³ − 23x² + 142x − 120 = (x − 1)(x − 10)(x − 12).

Question 5 (3 Marks - Identity-Based Evaluation)

Question: Without actual multiplication, calculate the value of (28)³ + (−15)³ + (−13)³ using algebraic identities.

Solution:

1. Let x = 28, y = −15, and z = −13.
2. Check the sum of the variables:
   x + y + z = 28 + (−15) + (−13) = 28 − 28 = 0
3. We know the conditional identity: If x + y + z = 0, then x³ + y³ + z³ = 3xyz.
4. Substitute the values:
   (28)³ + (−15)³ + (−13)³ = 3 × (28) × (−15) × (−13)
   = 3 × 28 × 195
   = 84 × 195 = 16,380

Answer: The calculated value is 16,380.

How to Prepare for CBSE Class 9 Polynomials

Scoring 100% in the algebra section of your board exam 9 syllabus requires methodical preparation. Follow these proven teacher-recommended strategies:

1. Master Identity Visualisation: Write all 8 algebraic identities on a formula sheet. Notice sign patterns in expansions like (x − y)³ versus (x + y)³ to prevent sign errors in calculations.
2. Maintain Rigorous Step Formatting: In CBSE marking schemes, marks are allotted for each step. Always write the general formula or theorem statement (e.g., "By Factor Theorem...") before performing algebraic substitutions.
3. Verify Factorisations by Expansion: After factoring expressions like (2x + 5)(3x + 1), quickly expand them in rough work to ensure you get back the original polynomial.
4. Target High-Weightage Problem Types: Spend extra time mastering cubic polynomial factorisation (rational root method + long division) and expressions involving x³ + y³ + z³ − 3xyz, as these consistently appear in 4-mark and 5-mark sections.

Where to Practice More

Consistent practice with chapter-wise questions, step-by-step verified keys, and timed mock tests is the most effective way to excel in CBSE Mathematics. Create custom practice sheets, download sample papers, and generate targeted question papers aligned with the latest CBSE pattern directly on Theorify QPTool (qptool.theorify.in) to accelerate your exam readiness today.