How do you factor 4x^3 + 36?

1 Answer
George C.
May 9, 2016

4x^3+36 = 4(x+root(3)(9))(x^2-root(3)(9)x+3root(3)(3))

Explanation:

First separate out the common scalar factor 4:

4x^3+36 = 4(x^3+9)

Note that x^3 is a perfect cube, but 9 is not - at least not a cube of a rational number.

We can still use the sum of cubes identity:

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

with a=x and b=root(3)(9) as follows:

x^3+9

=x^3+(root(3)(9))^3

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

=(x+root(3)(9))(x^2-root(3)(9)x+root(3)(81))

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

So:

4x^3+36 = 4(x+root(3)(9))(x^2-root(3)(9)x+3root(3)(3))

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