How do you factor ##x^3+343##?

##x^3+343=(x+7)(x^2-7x+49)##

The sum of cubes identity can be written:

##a^3+b^3 = (a+b)(a^2-ab+b^2)##

We can use this with ##a=x## and ##b=7## as follows:

##x^3+343##

##=x^3+7^3##

##=(x+7)(x^2-7x+7^2)##

##=(x+7)(x^2-7x+49)##

The remaining quadratic factor can only be factored further with Complex coefficients, which we can express in terms of the primitive Complex cube root of ##1##:

##=(x+7)(x+7omega)(x+7omega^2)##

where ##omega = -1/2+sqrt(3)/2i##