How do you factor x^3+27?

2 Answers
Nov 3, 2016

Because factoring x^3+27 is the same as finding where the graph passes through the x axis, we can just set the equation equal to zero and solve. f(x) = (x+3)(x^2-3x+9)

Explanation:

Let f(x) = x³ + 27
0 = x³ + 27
x³= -27
x = -3
This means that x = -3 is the only zero of the graph of f(x). Since we know (x+3) is one factor of f(x), to find the 2nd factor,
(x^3+27)/(x+3)
We get the second factor to be x^2-3x+9.

Therefore, f(x) = (x+3)(x^2-3x+9)

Nov 3, 2016

x^3+27 = (x+3)(x^2-3x+9)

Explanation:

The sum of cubes identity can be written:

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

Note that both x^3 and 27=3^3 are perfect cubes, so we can use the sum of cubes identity with a=x and b=3 as follows:

x^3+27 = x^3+3^3

color(white)(x^3+27) = (x+3)(x^2-3x+3^2)

color(white)(x^3+27) = (x+3)(x^2-3x+9)

The remaining quadratic has no simpler linear factors with Real coefficients.