How do you factor the expression #x^3+1/8 #?

1 Answer
sente
Dec 5, 2015

#x^3 + 1/8 = (x + 1/2)(x^2 -1/2x + 1/4)#

Explanation:

The sum of cubes formula states that

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

(Verify this by multiplying it out)

As #1/8 = (1/2)^3# we can apply the formula.

#x^3 + 1/8 = x^3 + (1/2)^3 = (x + 1/2)(x^2 -1/2x + 1/4)#

Note that we cannot factor any further, as the discriminant for the remaining quadratic is negative:
#(-1/2)^2 - 4(1)(1/4) = 1/4 - 1 = -3/4#
and thus it has no real roots.

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