can you give me detailed list and explanation of all the identities of polynomials fast - Maths - Polynomials

Question

can you give me detailed list and explanation of all the
identities of polynomials fast

Thundersam · Accepted Answer

There are 8 important algebraic identities which are given
below:
POLYNOMIALS OF DIFFERENT DEGREES:
Identity I:
(x + y)2= x2+ 2xy + y2
Identity II:
(x - y)2= x2- 2xy + y2
Identity III:
x2- y2= (x+ y)(x - y)
Identity IV:
(x + a)(x + b) = x2+ (a + b)x + ab
Identity V:
(x + y + z)2= x2+ y2+
z2+ 2xy + 2yz + 2zx
Proof:Let x + y = k then,(x + y +
z)2= (k + z)2= k2+ 2kz +
z2(Using identity I)= (x + y)2+ 2(x + y)z +
z2= x2+ 2xy + y2+ 2 xz + 2yz +
z2= x2+ y2+ z2+ 2xy +
2yz + 2zx (proved)
Identity VI:
(x + y)3= x3+ y3+ 3xy(x +
y)
Identity VII:
(x - y)3= x3- y3- 3xy(x -
y)
Identity VIII:
x3+ y3+ z3- 3xyz = (x + y +
z)(x2+ y2+ z2- xy - yz - zx)
Proof:R H S = (x + y + z)(x2+
y2+ z2- xy - yz - zx) = x(x2+
y2+ z2- xy - yz - zx) + y(x2+
y2+ z2- xy - yz - zx) + z(x2+
y2+ z2- xy - yz - zx) = x3+
xy2+ xz2- x2y - xyz -
zx2+ yx2+ y3+ yz2-
xy2- y2z - xyz + zx2+
zy2+ z3- xyz - yz2-
xz2= x3+ y3+ z3- 3xyz =
L H S (proved)
Example 1:Evaluate 95 x 96
Solution:95 x 96 = (90 + 5) x (90 + 6) =
(90)2+ (5 + 6)(90) + (5)(6) (Using identity IV) = 8100 +
990 + 30 = 9120(Answer)
Example 2:Evaluate 104 x 96
Solution:104 x 96 = (100 + 4) x (100 - 4) =
(100)2- (4)2(Using identity III) = 10000 - 16
= 9984(Answer)
Example 3:Factorise 4x2+ 2xy +
y2 Solution:4x2+ 2xy +
y2= (2x)2+ 2(2x)(y) + (y)2= (2x +
y)2(Using identity I) = (2x + y)(2x +
y)(Answer)
Example 4:Factorise 9x2- 6xy +
y2 Solution:9x2- 6xy +
y2= (3x)2- 2(3x)(y) + (y)2= (3x -
y)2(Using identity II) = (3x - y) (3x -
y)(Answer)
Example 5:Expand (3x - 7y - z)2
Solution:(3x - 7y - z)2=
{3x+(-7y)+(-z)}2(Using identity V) = (3x)2+
(-7y)2+ (-z)2+ 2(3x)(-7y) + 2(-7y)(-z) +
2(-z)(3x) = 9x2+ 49y2+ z2- 42xy +
14yz - 6zx(Answer)
Example 6:Expand (2x + 1)3
Solution:(2x + 1)3= (2x)3+
(1)3+ 3(2x)(1)(2x + 1) (Using identity VI) =
8x3+ 1 + 6x(2x + 1) = 8x3+ 12x2+
6x + 1(Answer)
Example 7:Expand (2x - 3y)3
Solution:(2x - 3y)3= (2x)3-
(3y)3- 3(2x)(3y)(2x - 3y) (Using identity IV) =
8x3- 27y3- 18xy(2x - 3y) = 8x3-
27y3- 36x2y +
54xy2(Answer)
Example 8:Factorise 27x3+
y3+ z3- 9xyz
Solution:27x3+ y3+
z3- 9xyz = (3x)3+ (y)3+
(z)3- 3(3x)(y)(z) = (3x + y + z){(3x)2+
(y)2+ (z)2- (3x)(y) - (y)(z) - (z)(3x)}
(Using identity VIII) = (3x + y + z)(9x2+ y2+
z2- 3xy - yz - 3zx)(Answer)Remember
a3+b3can also be written as
(a+b)3-3ab(a+b)e g a+b =4 then find the value of
a3+b3+12ab-64 try it by using above
formula
HERE it is, now study hard

can you give me detailed list and explanation of all the identities of polynomials.....fast....