Question

How do you factor `a^6 + 1`?

Answer 1

We can modify the expression to use the Sum of Cubes formula to factorise it.

`a^6 + 1 = (a^2)^3 + 1^3`

The formula says :`color(blue)(x^3 + y^3 = (x + y)(x^2-xy+y^2)`

Here, `x` is `a^2` and `y` is `1`

`(a^2)^3 + 1^3 = (a^2+1){(a^2)^2 - (a^2*1)+1^2}`

`color(green)(= (a^2+1)(a^4 - a^2+1)`

As none of the factors can be factorised further, this becomes the Factorised form of `a^6 + 1`