Question

How do you factor `8k^3 + 1`?

Answer 1

`(2k+1)(4k^2-2k+1)`

Explanation:

This is the factorization of perfect cubes.

The pattern for factoring perfect cubes is
`(a+b)(a^2-ab+b^2)`
or
`(a-b)(a^2+ab+b^2)`
The pattern is dependent upon the sign between the cubes.

a = the cubic factor of the first term
b = the cubic root of the second term

(first term, second term) (first term squared, first term x second term, second term squared.

The pattern of the pattern of the signs follows the SOAP rule
S - Same Sign
O - Opposite Sign
AP - Always Positive

(Same Sign) (Opposite Sign, Always Positive)

`8k^3 + 1`
The cubic factor of `8k^3` is 2k
The cubic factor of 1 is 1

`a = 2k`
`b = 1`

`(2k + 1)((2k)^2-(2k)(1)+(1)^2)`
`(2k + 1)(4k^2-2k+1)`