Find the complex zeros

Question

Find the complex zeros of the following polynomial function. Write f in factored for f(x)

 

 

f (x)= x^3+ 1

a. The complex zeros of f are?????

 

b. Use the complex zeros to factor f.

Answer 1

f(x) = x³  + 1

 

This is really a sum of cubes, f(x) = x³ + 1³.  There is a standard factoring formula for a sum of cubes:

 

a³ + b³ = (a + b)(a² – ab + b²)

 

In our case, a = x and b = 1, so:

 

x³ + 1³ = (x+1)(x² - x + 1)

 

The real root, then, is x = -1.  To find the complex roots, apply the quadratic formula to the quadratic expression in the parentheses, x² - x + 1:

 

x = -b/2a ± (√(b²-4ac))/2a

x = 1/2 ± (√(1-4·1·1))/2

x = 1/2 ± i(√3)/2    ---->  These are the complex zeros

 

Fully factored:

 

f(x) = (x + 1)(x + (1/2 + i(√3)/2)(x + (1/2 - i(√3)/2)