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Answers
Algebraic Identities
Polynomials can be factorised using algebraic identities.
A polynomial of degree two is called a quadratic polynomial. The identities used to factorise the quadratic polynomials are:
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a2 – b2 = (a + b)(a – b)
(x + a)(x + b) = x² + (a + b)x + ab
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
A polynomial of degree three is called a cubic polynomial. The algebraic identities used in factorising a cubic polynomial are:
(a + b)³ = a³ + b³ + 3ab (a + b)
(a – b)³ = a³ – b³ – 3ab (a – b)
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)
Answer:
Important Polynomial Identities :
1) ( x + y )2 = x2 + 2xy +y2
2) ( x – y) 2 = x2 – 2xy + y2
3) (x + y)(x – y) = x2 – y2
4) (x + a)(x + b) = x2 +(a + b)x + ab
5) (x + y) 3 = x3 + 3x2y + 3xy2 + y3 = x3 + y3 +3xy(x +y)
6) (x - y) 3 = x3 - 3x2y + 3xy2 - y3 = x3+ y3 -3xy(x –y)
7) (x + y + z) 2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
8) x3 + y3 = (x + y)(x2 – xy + y2)
9) x3 - y3 = (x - y)(x2 + xy + y2)
10) x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)
If x + y + z = 0 , then x3 + y3 + z3 = 3xyz