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:
当b =,
(x + a) (x + a) = x2+ 2ax + a2
When = - a, we have = x2- a2When 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+ b3To 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- 3 abc。
(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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