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Factoring using formulas
Some algebraic expressions can be readily factored by using formulas
Frequently used formulas for factoring are listed as follows:
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
Proof
used to factor perfect square trinomials
a2 - b2 = (a + b)(a - b)
Proof
used to factor the difference of two squares.
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
used to factor the sum or difference of two cubes
a3 + 3a2b + 3ab2 + b3 = (a + b)3
a3 - 3a2b + 3ab2 - b3 = (a -
b)3
Proof
used to factor the perfect cube polynomials
an - bn = (a - b)(an - 1 + an - 2b
+ an - 3b2........ + a2bn - 3+ abn
- 2 + bn - 1)
for n > 1
(Note that
a2 - b2 = (a + b)(a - b) is a special case of the formula
where n = 2, and
a3 - b3 = (a - b)(a2 + ab + b2) is a
special case of the formula when n = 3.)
In a special case where b = 1, we have
an - 1 = (a - 1)(an - 1 + an - 2 + an -
3........ + a2+ a + 1)
an + bn = (a + b)(an - 1 - an - 2b
+ an - 3b2........ + a2bn - 3 - abn
- 2 + bn - 1)
where n is an odd natural number and n > 1
(Note that
a3 + b3 = (a + b)(a2 - ab + b2) is a
special case of the formula when n = 3.)
a2 + b2 + c2 + 2ab + 2ac + 2bc = (a + b + c)2 Proof
To learn how to use the formulas to factor algebraic expressions, please click the following links.
Factoring perfect square trinomials
Factoring the difference of two squares
Factoring the sum or difference of two cubes
Factoring the perfect cube polynomials
Factoring the difference of two powers an - bn
Factoring the sum of two powers an + bn, where n is a odd natural number greater than 1.