27x3 + y3 + z3 - 9xyz
[∵ x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)]
= (3x)3 + y3 + z3 - 3(3x)(y)(z)
= (3x + y + z)[(3x)2 + y2 + z2 - (3x)(y) - (y)(z) - (z)(3x)]
= (3x + y + z)(9x2 + y2 + z2 - 3xy - yz - 3zx)
Hence, 27x3 + y3 + z3 - 9xyz = (3x + y + z)(9x2 + y2 + z2 - 3xy - yz - 3zx)
BRIGHT TUTORIALS
BRIGHT TUTORIALS
CBSE Class IX | Academic Year 2026-2027
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Excellence in Education
Mathematics | PolynomialsWeb Content • Interactive Notes
Polynomials — Interactive Study Guide
Master polynomial basics, Remainder and Factor Theorems, factorisation, and algebraic identities.
Polynomial Classification
Not a polynomial: √x (fractional power), 1/x = x−1 (negative power), x + 1/x.
Polynomial: 5 (constant), 3x + 2 (linear), x² − 1 (quadratic), 2x³ + x − 1 (cubic).
Remainder and Factor Theorems — Quick Guide
Remainder Theorem: When p(x) is divided by (x − a), remainder = p(a).
Factor Theorem: (x − a) is a factor of p(x) ⇔ p(a) = 0.
Factor Theorem: (x − a) is a factor of p(x) ⇔ p(a) = 0.
Watch the sign! Dividing by (x + 3) means a = −3. So remainder = p(−3).
Identity Mastery Checklist
| See This Pattern | Use This Identity |
|---|---|
| a² + 2ab + b² | = (a + b)² |
| a² − 2ab + b² | = (a − b)² |
| a² − b² | = (a + b)(a − b) |
| a³ + b³ | = (a + b)(a² − ab + b²) |
| a³ − b³ | = (a − b)(a² + ab + b²) |
| a + b + c = 0 | ⇒ a³ + b³ + c³ = 3abc |
Quick Self-Check
- Degree of 5x³ − 2x + 1? (3)
- Remainder when x² + 3x + 2 is divided by (x + 1)? (p(−1) = 1 − 3 + 2 = 0)
- Expand: (2a + 3b)² (= 4a² + 12ab + 9b²)
- Factorise: 8x³ − 27 (= (2x − 3)(4x² + 6x + 9))