Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Quick Summary
Algebra in Class 9 focuses on polynomials — degree, types, operations, division, and theorems. The Factor Theorem and Remainder Theorem are the most tested concepts. Algebraic identities (especially cubic identities) are frequently asked in 3-mark and 5-mark questions.
Key Definitions
- Polynomial: An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ where all exponents are non-negative integers.
- Degree: Highest power of the variable with a non-zero coefficient.
- Zero of a polynomial: Value of x for which p(x) = 0.
- Monomial: One term (e.g., 3x²).
- Binomial: Two terms (e.g., x + 5).
- Trinomial: Three terms (e.g., x² + 3x + 2).
Types of Polynomials by Degree
| Name | Degree | Example |
|---|---|---|
| Constant | 0 | 7 |
| Linear | 1 | 2x + 3 |
| Quadratic | 2 | x² − 5x + 6 |
| Cubic | 3 | 2x³ + x − 1 |
| Biquadratic | 4 | x⁴ − 1 |
Division Algorithm for Polynomials
Dividend = Divisor × Quotient + Remainder
Steps: 1. Arrange both polynomials in descending order of degree. 2. Divide the leading term of dividend by leading term of divisor. 3. Multiply the divisor by the result and subtract. 4. Repeat until the degree of remainder < degree of divisor.
Remainder Theorem
When p(x) is divided by (x − a): Remainder = p(a)
Factor Theorem
(x − a) is a factor of p(x) if and only if p(a) = 0
Key Algebraic Identities
- (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)
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
- a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
Important Questions
One Mark
1. Find the degree of 5x³ − 2x + 7. 2. If p(x) = x² − 3x + 2, find p(1). 3. State the Remainder Theorem. 4. Is (x − 1) a factor of x³ − 1? Verify using Factor Theorem. 5. Write the zero of the polynomial 2x + 6.
Short Answer — 3 marks
1. Find the remainder when 2x³ − 3x² + 4x − 5 is divided by (x − 2) using Remainder Theorem. 2. Factorise: x³ − 6x² + 11x − 6 using Factor Theorem. 3. Expand (2x + 3y)³ using the identity.
Long Answer — 5 marks
1. Divide 6x³ − 13x² + x + 2 by (2x + 1) using long division. State the quotient and remainder. 2. If (x − 2) and (x + 1) are factors of p(x) = x³ + ax² + bx + c and the remainder when divided by (x − 3) is 6, find a, b, and c.
Common Mistakes
- Including terms with negative exponents (e.g., x⁻¹) as polynomials — they are NOT polynomials.
- Confusing degree with leading coefficient.
- Sign errors when subtracting in long division — always change all signs carefully.
- Forgetting that Factor Theorem requires p(a) = 0, not p(−a) = 0 for (x − a).
Revision Checklist
- [ ] Know types of polynomials by degree with examples
- [ ] Can find zeroes of linear and simple quadratic polynomials
- [ ] Can apply Remainder Theorem correctly
- [ ] Can use Factor Theorem to factorise cubic polynomials
- [ ] Memorised all four cubic identities
- [ ] Can perform polynomial long division