How do you factor x cubed plus one?

Answer 1

Whenever you want to factorize a polynomial F(x), it is useful if you can see a value of x that makes F(x) = 0. If you find one (call it "a") then (x-a) will be a factor of F(x).

Therefore start by noticing that the polynomial x3+1 is zero when x = -1

This means that (x + 1) is a factor, so

x3+1 = (x + 1)(ax2+bx+c)

for some values of coefficients a, b and c.

Multiply out the brackets and collect terms:

x3+1=ax3+(a+b)x2+(b+c)x+c

From coefficients of x3: a = 1

From coefficients of x2: b = -1

From coefficients of x: c = 1

So x3+1 = (x+1)(x2-x+1)

To factorise further we need to solve the quadratic equation

x2-x+1 = 0

x2-x+1 = (x - (1/2))2+(3/4) = 0

(x - (1/2))2 = -3/4

The solutions of this are

x = (1/2) + i(3(1/2))/2

and

x = (1/2) - i(3(1/2))/2

where "3(1/2)" means the square root of 3

Call these values w and z.

Then we have finished:

x3+1 = (x + 1)(x - w)(x - z)

But if you don't want an answer with "i" in it, the best you can do is

x3+1 = (x + 1)(x2-x+1)