R.H.S
= (x + y + z)[(x - y)2 + (y - z)2 + (z - x)2]
= (x + y + z)[(x2 + y2 - 2xy) + (y2 + z2 - 2yz) + (z2 + x2 - 2zx)] [∵ (a - b)2 = (a)2 + (b)2 - 2ab]
= (x + y + z)[2x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx]
= (x + y + z) 2[x2 + y2 + z2 - xy - yz - zx]
= (x + y + z)[x2 + y2 + z2 - xy - yz - zx]
= x3 + y3 +z3 - 3xyz [∵ x3 + y3 +z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)]
Hence, L.H.S = R.H.S
BRIGHT TUTORIALS
BRIGHT TUTORIALS
CBSE Class IX | Academic Year 2026-2027
9403781999
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))