Identities, Cube of a binomial, Algebra, Math Assignment Help

mathematics - Identities, Cube of a binomial, Algebra, Math

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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