any four identities of polynomial
Answers
Answer: Polynomial Identities : An algebraic expression in which the variables involved have only non negative integral powers is called polynomial.
For factorization or for the expansion of polynomial we use the following identities.
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
Step-by-step explanation
Some solved examples :
Expand the following using Polynomial Identities.
1) (2a + 5) 2
Solution : (2x + 5) 2 = (2a) 2 + 2(2a)(5) + 5 2
[ using the identity ( x + y ) 2 = x 2 + 2xy +y 2 ]
= 4a 2 + 20 a + 25
_________________________________________________________________
2) ( b + 6)(b - 6)
[ using the identity (x + y)(x – y) = x 2 – y 2 ]
( b + 6)(b - 6) = b 2 - 6 2
= b 2 - 36
__________________________________________________________________
3) ( 3a - 4) 3
[using the identity(x - y) 3 = x 3 - 3x 2 y + 3xy 2 - y 3 ]
( 3a - 4) 3 = (3a) 3 - (3a) 2 (4) + 3(3a)(4) 2 - 4 3
= 27a 3 -72a 2 + 36a - 64
__________________________________________________________________
Factorize the following using Polynomial Identities :
1) Factorize: 64a 3 - 27b 3 - 144a 2 b + 108ab 2 .
Solution :
64a 3 - 27b 3 - 144a 2 b + 108ab 2
= (4a) 3 - (3b) 3 - 36ab(4a- 3b)
= (4a) 3 - (3b) 3 -3(4a)(3b)(4a -3b)
= (4a - 3b) 3 [ using x 3 + y 3 -3xy(x –y) ]
= (4a -3b)(4a -3b)(4a -3b)
_________________________________________________________________
2)Evaluate : (104) 3
(104) 3 = ( 100 + 4) 3
= = (100) 3 + (4) 3 + 3(100)(4)(100 + 4) [Using Identity V]
= 1000000 + 64 + 124800
= 1124864
_______________________________________________________________
3) Evaluate : (–12) 3 + (7) 3 + (5) 3
Solution : (–12) 3 + (7) 3 + (5) 3
From the above we can see that -12 + 7 + 5 = 0
(–12) 3 + (7) 3 + (5) 3 = 3(-12)(7)(5) [Using identity 10]
= -1260
Answer:
The four identities of polynomial are:-
(i)( x + y ) 2 = x 2 + 2xy +y 2
(ii) ( x – y) 2 = x 2 – 2xy + y 2
(iii) (x + y)(x – y) = x 2 – y 2
(iv) (x + a)(x + b) = x 2 +(a + b)x + ab.