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Expansions, Expressions and Identities

PRODUCTS


We get one expression when two or more expressions are multiplied. The result is universally true and then it is called an identity. These identities can be used to reduce large products into simpler ones. For example (x + a) is one expression and (x - a) is another expression and when they are multiplied we get x2 - a2. (x + a)(x - a) = x2 - a2 is an identity. Let us look at some of the products.

(x + a)(x + b) = x2 + (a + b)x + ab
(a + b)(a - b)= a2 - b2
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

Product of two expressions (x + a) and (x + b)

(x + a)(x + b)
= x(x + b) + a(x + b)
= x2 + bx + ax + ab
= x2 + (a + b)x + ab

Product of sum and difference of two terms a and b

(a + b)(a - b)
= a2 + ba - ba - b2
= a2 - b2

Square of binomials

(a + b)2
= (a + b)(a + b)
= a2 + 2ab + b2

(a - b)2
= (a - b)(a - b)
= a2 - 2ab + b2

Square of trinomials

(a + b + c)2
= (a + b + c)(a + b + c)
= (a(a + b + c) + b(a + b + c) + c(a + b + c))
= (a2 + ab + ac + ba + b2 + bc + ca + cb + c2)
= a2 + b2 + c2 + 2ab + 2bc + 2ca


Important Points

  1. Product of two binomials

    (x + a)(x + b) = x2 + (a+b)x + ab
    (x + a)(x - a) = x2 - a2
  2. Square of a binomial

    (a + b)2 = a2 + 2ab + b2
  3. Square of a binomial

    (a − b)2 = a2 −2ab + b2
  4. Square of a trinomial

    (a + b + c)2 = a2 + b2 + c2 + 2( ab + bc + ca)
  5. Cube of a binomial

    (a + b)3 = a3 + b3 + 3ab(a + b)
  6. Cube of a binomial

    (a − b)3 = a3 + b3 − 3ab(a + b)

(a ± b)2 = a2 ± 2ab + b2

a2 + b2 = (a + b)2 − 2ab
a2 + b2 = (a − b)2 + 2ab
(a + b)2 = (a − b)2 + 4ab
(a − b)2 = (a + b)2 − 4ab
(a + b)2 + (a − b)2 = 2(a2 + b2)
(a + b)2 − (a − b)2 = 4ab



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