NCERT Class 9 Polynomials- Complete Study Guide
NCERT Class 9 Polynomials: What You Actually Need to Know
Polynomials are one of the most important chapters in Class 9 Maths. Most students either master it completely or lose marks unnecessarily. There's no middle ground because the concepts build directly on what comes next in Class 10 and beyond.
This guide cuts through the textbook fluff. You'll find exactly what matters, what to practice, and where most students go wrong.
What Is a Polynomial?
A polynomial is an algebraic expression with variables and coefficients, combined using addition, subtraction, and multiplication. The exponents must be whole numbers—no fractions, no negatives.
Examples:
- 3x² + 5x - 2
- 4y³ - 7y + 1
- 5 (this is a constant polynomial)
Non-examples:
- x² + 1/x (negative exponent)
- √x + 3 (not a whole number exponent)
Key Terms You Must Know
Variable: The letter part—x, y, z. Nothing fancy here.
Coefficient: The number multiplied by the variable. In 4x³, the coefficient is 4.
Term: Each part separated by + or - signs. 3x² has one term.
Degree: The highest exponent in the polynomial. This matters more than most students realize.
Types of Polynomials
Based on the number of terms:
- Monomial: One term. Example: 5x³
- Binomial: Two terms. Example: x + 1
- Trinomial: Three terms. Example: x² + 2x + 1
Based on degree:
- Linear: Degree 1. Form: ax + b. Example: 3x + 5
- Quadratic: Degree 2. Form: ax² + bx + c. Example: 2x² + 3x - 1
- Cubic: Degree 3. Form: ax³ + bx² + cx + d. Example: x³ - 4x² + 2x + 7
The Zeroes of a Polynomial
The zeroes of a polynomial p(x) are the values of x where p(x) = 0. Simple to understand, but students mess up the calculations.
Example: Find the zeroes of p(x) = x² - 5x + 6
Set p(x) = 0:
x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3
The zeroes are 2 and 3. Verify: p(2) = 4 - 10 + 6 = 0 ✓
The Remainder Theorem
Statement: When you divide a polynomial p(x) by (x - a), the remainder is p(a).
This theorem saves you from doing long division when you only need the remainder.
Example: Find the remainder when p(x) = x³ + 2x² - 5x + 1 is divided by (x - 2).
According to the theorem, remainder = p(2)
p(2) = (2)³ + 2(2)² - 5(2) + 1
p(2) = 8 + 8 - 10 + 1
p(2) = 7
The remainder is 7. No division required.
The Factor Theorem
Statement: If p(a) = 0, then (x - a) is a factor of p(x). Conversely, if (x - a) is a factor, then p(a) = 0.
This is the Remainder Theorem's reverse. Use it to check if something is a factor or to find unknown coefficients.
Example: Check if (x - 1) is a factor of p(x) = x² - 3x + 2
p(1) = (1)² - 3(1) + 2 = 1 - 3 + 2 = 0
Since p(1) = 0, (x - 1) is a factor. ✓
Algebraic Identities You Must Memorize
These four identities appear in almost every polynomial problem. Memorize them before your exam.
| Identity | Formula |
|---|---|
| Square of Binomial | (a + b)² = a² + 2ab + b² |
| Square of Binomial | (a - b)² = a² - 2ab + b² |
| Difference of Squares | (a + b)(a - b) = a² - b² |
| Product of Binomials | (x + a)(x + b) = x² + (a + b)x + ab |
Advanced Identities for Higher Marks
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a - b)³ = a³ - 3a²b + 3ab² - b³
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
How to Solve Polynomial Problems: A Practical Approach
Step 1: Identify the Question Type
Are you finding zeroes? Verifying an identity? Finding a remainder? Know what you're solving before you start.
Step 2: Write Down Given Information
Note the polynomial, the divisor or value, and what you need to find. Skipping this step causes careless errors.
Step 3: Apply the Relevant Theorem
- Remainder only → Use Remainder Theorem
- Checking factors → Use Factor Theorem
- Expanding expressions → Use algebraic identities
- Factorizing → Look for common factors, then try splitting or identities
Step 4: Verify Your Answer
Check by substitution. If you found zeroes, verify each one. If you factored something, multiply back to confirm.
Common Mistakes Students Make
- Confusing degree with number of terms. Degree is the highest exponent. A cubic polynomial has degree 3, not 3 terms.
- Forgetting to check all conditions before applying the Factor Theorem. p(a) must equal 0.
- Misremembering identities. The signs in (a - b)² and (a - b)³ trip up many students.
- Not simplifying completely. Always combine like terms before moving to the next step.
- Ignoring constant polynomials. A constant like 5 is still a polynomial—just with degree 0.
How to Practice Effectively
Don't just read solutions. Attempt problems yourself first.
Start with NCERT examples, then move to exercises. The NCERT textbook has enough problems to build solid understanding if you solve them genuinely.
Focus on factorization problems until you can do them without hesitation. They're the foundation for Class 10 quadratic equations.
Time yourself during revision. Polynomial problems shouldn't take more than 5-7 minutes in an exam if you know your concepts.
What Comes Next
After polynomials, coordinate geometry and geometry follow in Class 9. But polynomials connect directly to Class 10's quadratic equations and arithmetic progressions.
If your basics here are weak, you'll struggle later. Fix it now.