Identity: An identity is an equality which is true for all values of variables.
IMPORTANT FORMULAE
(1) To Find the Product (x + a) (x + b): (x + a) (x + b)
= x2 + (a + b)x + ab
Using this identity, we can derive the following results:
When b = a, we have
(x + a) (x + a) = x2 + 2ax + a2
When = - a, we have = x2 - a2 When b = - b, we have
= x2 + (a - b)x - ab
When a = - a, we have = x2 + (b - a) x - ab
When a = - a, b = - b, we have = x2 - (a + b) x + ab.
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b) (a - b) = a2 - b2.
To Find the Product (a + b) (a2 - ab + b2): (a + b) (a2 - ab + b2)
= a3 + b3 To Find the Product (a - b)(a2 + ab + b2) :
(a - b) (a2 + ab + b2) = a3 - b3
To Find the Product (a + b + c) (a2 + b2 + c2 - ab - bc - ac):
(a + b + c) (a2 + b2 + c2 - ab - bc - ac) = a3 + b3 + c3 - 3abc.
(2) Cube of a binomial:
(a + b)3 = a3 + b3 + 3ab (a + b)
a3 + b3 = (a + b)3 - 3ab (a + b)
(a - b)3 = a3 - b3 - 3ab (a - b)
a3 - b3 = (a - b)3 + 3ab (a - b).
(3) Square of a trinomial:
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac.
Note: a, b, c may take various positive and negative values:
(i) (- a - b + c)2 = a2 + b2 + c2 + 2ab - 2ac - 2bc.
(ii) (a - b + c)2 = a2 + b2 + c2 - 2ab - 2bc + 2ac.
(iii) (- a + b + c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ac.
(iv) (a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc + 2ac.
(v) (a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ac.
An important result: If a + b + c = 0, then a3 + b3 + c3 = 3abc.
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