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

1 Answer
Apr 23, 2016

Answer:

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

Explanation:

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#