(i) 27y3 + 125z3
[∵ x3 + y3 = (x + y)(x2 - xy + y2)]
⇒ 27y3 + 125z3 = (3y)3 + (5z)3
Putting x = 3y, y = 5z
R.H.S = (x + y)(x2 - xy + y2)
= (3y + 5z)[(3y)2 - (3y)(5z) + (5z)2]
= (3y + 5z)(9y2 - 15yz + 25z2)
Hence, 27y3 + 125z3 = (3y + 5z)(9y2 - 15yz + 25z2)
(ii) 64m3 - 343n3
[∵ a3 - b3 = (a - b)(a2 + ab + b2)]
⇒ 64m3 - 343n3 = (4m)3 - (7n)3
Putting a = 4m, b = 7n
R.H.S = (a - b)(a2 + ab + b2)
= (4m - 7n)[(4m)2 + (4m)(7n) + (7n)2]
= (4m - 7n)(16m2 + 28mn + 49n2)
Hence, 64m3 - 343n3 = (4m - 7n)(16m2 + 28mn + 49n2)
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
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))