Question

Factor the polynomial

X^3 + 8

Answer 1

google "sum of cubes"

x^3 + 2^3

Answer 2

To factor x³ + 8 you must use sum of cubes formula:

A³ + B³ = ( A + B ) ( A² - A ∙ B + B² )

In this case :

A = x

B = 2

becouse 2³ = 8

So:

x³ + 8 = x³ + 2³

x³ + 2³ = ( x + 2 ) ( x² - x ∙ 2 + 2² ) =

( x + 2 ) ( x² - 2 x + 4 )


x² - 2 x + 4 can't be factoring becouse discriminant of x² - 2 x + 4:

D = b² - 4 ∙ c = ( - 2 )² - 4 ∙ 1 ∙ 4 = 4 - 16 = - 12

This mean:

x² - 2 x + 4 has two complex solutions and can't be factoring with real numbers.


Solution:

x³ + 8 = ( x + 2 ) ( x² - 2 x + 4 )