Polynomials — Class 9 Foundation
Polynomials is one of the most important Class 9 chapters for JEE preparation. The Remainder Theorem, Factor Theorem, and Algebraic Identities introduced here are used throughout JEE Algebra — in Quadratic Equations, Complex Numbers, and Sequences. Building a strong foundation now saves significant time later.
Prerequisites: Basic arithmetic and algebraic expressions
Key Concepts
What you need to master:
- Degree of a polynomial, types (linear, quadratic, cubic)
- Remainder Theorem: remainder when p(x) is divided by (x-a) is p(a)
- Factor Theorem: (x-a) is a factor of p(x) iff p(a) = 0
- Factorization of polynomials using factor theorem
- Algebraic identities: (a+b)², (a-b)², a²-b², (a+b)³, (a-b)³, a³+b³, a³-b³
- Factorization using identities
Remainder and Factor Theorems
Remainder Theorem: if you divide polynomial p(x) by (x - a), the remainder is simply p(a). No long division needed.
Factor Theorem: if p(a) = 0, then (x - a) is a factor of p(x). This is the primary method for factoring cubics and higher-degree polynomials — guess a root using trial (try ±1, ±2, etc.), confirm with factor theorem, then divide.
These theorems are the foundation for solving polynomial equations in JEE. In Class 11-12, they extend to finding roots of higher-degree equations and connecting to Vieta's formulas.
Algebraic Identities for JEE
(a+b)² = a² + 2ab + b². (a-b)² = a² - 2ab + b². a² - b² = (a+b)(a-b).
(a+b)³ = a³ + 3a²b + 3ab² + b³. a³ + b³ = (a+b)(a² - ab + b²). a³ - b³ = (a-b)(a² + ab + b²).
These identities appear everywhere in JEE — in simplifying expressions, in Sequences & Series, and even in Integration (partial fractions). Memorize them so thoroughly that you can apply them without thinking.
Key Formulas
Common Mistakes to Avoid
- Mistake: Confusing remainder theorem (dividing by x-a gives remainder p(a)) with dividing by x+a (remainder is p(-a))
- Mistake: Sign errors in (a-b)² and (a-b)³ expansions
- Mistake: Forgetting to check all possible rational roots (± factors of constant term) when factoring
- Mistake: Not verifying factorization by expanding back
How to Practice This Topic
Do all NCERT exercises. Memorize all identities by writing them out daily for a week. Practice factoring 10 cubics using the factor theorem. These skills are used in every JEE Algebra topic.
Formula Sheets
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