Factoring Third Degree Polynomials

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To factor the third degree polynomial, the following formulae can be used.

(a) a3 + b3 =   (a + b) (a2 – ab + b2)
(b) a3 - b3  =   (a - b) (a2 + ab + b2)
(c) (a + b)3 =   a3 + 3 a2b + 3 ab2 + b3
(d) (a - b)3 =   a3 - 3 a2b + 3 ab2 - b3

Example : Let us factorize the third degree polynomial 125 x3 - 1.
Solution : The given third degree polynomial is 125 x3 - 1.
It can be written as
125 x3 – 1 = (5x)3 - 1
a3 - b3   =   (a - b) (a2 + ab + b2)
(5x)3 – 1 =  ( 5x -1) ( 25 x²  + 5x + 1)

Example : Let us factorize the third degree polynomial 625 x3 + 216.
Solution : The given third degree polynomial is15625 x3 + 216.
It can be written as
15625 x3 + 216 = (25x)3 + 63
a3 + b3 =   (a + b) (a2 – ab + b2)
(25x)3 + 63 =  ( 25x + 6) ( 625 x²  - 150 x + 36)

Example : Let us factorize the third degree polynomial a3 + 3 a2b + 3 ab2 + b3
Solution : The given third degree polynomial is a3 + 3 a2b + 3 ab2 + b3
It can be written as
a3 + 3 a2b + 3 ab2 + b3
= (a + b) (a² + 2ab + b²)
= (a + b) (a + b) ²
= (a + b) ³

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