8
CBSE Marks
★★★★★
Difficulty
8
Topics
High
Board Weight
Topics Covered
8 key topics in this chapter
Polynomials & Their Degrees
Types: Monomial, Binomial, Trinomial
Value & Zeros of a Polynomial
Remainder Theorem
Factor Theorem
Factorisation using Identities
Algebraic Identities (5 key)
Division of Polynomials
Study Resources
Key Formulas & Identities
| Formula / Rule | Expression |
|---|---|
| (a+b)² | \(a² + 2ab + b²\) |
| (a−b)² | \(a² − 2ab + b²\) |
| a²−b² | \((a+b)(a−b)\) |
| (a+b)³ | \(a³ + 3a²b + 3ab² + b³\) |
| (a−b)³ | \(a³ − 3a²b + 3ab² − b³\) |
| a³+b³ | \((a+b)(a²−ab+b²)\) |
| a³−b³ | \((a−b)(a²+ab+b²)\) |
| a³+b³+c³−3abc | \((a+b+c)(a²+b²+c²−ab−bc−ca)\) |
| Special case | \(a+b+c=0 ⟹ a³+b³+c³ = 3abc\) |
Important Points to Remember
Remainder Theorem: when p(x) is divided by (x − a), the remainder is p(a). This avoids long division for simple checks.
Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. The converse is equally important.
Degree of the zero polynomial is not defined. A non-zero constant polynomial has degree 0.
Key identities to memorise: (a+b)² = a²+2ab+b²; (a−b)² = a²−2ab+b²; a²−b² = (a+b)(a−b); (a+b)³; (a−b)³; a³+b³+c³−3abc.