Factor the following sum of two cubes.

Question

z^3 + 125

Select the correct choice below.

(a) z^3 + 125=? (Factor completely. Simplify your answer.)

(b) The polynomial is prime.

 

 

Answer 1

Hi Jamal,

To factor this completely, you start by determining the first, most apparent factor, which you get by setting the expression equal to zero and solving for z.  

You get z^3 = -125   ------>>   z = -5

Therefore, one of your factors is (z+5).  

To get the other two, you need to divide this factor into the expression, so you set up the synthetic division like this:

-5    1  0  0     125                

          -5  25  -125

       1   -5  -125   0 

(The -5 is what you're dividing by. The other numbers are the coefficients of the 3rd order polynomial, with zeros in place for the missing x^2 and x terms.  Simply bring down the first coefficient unchanged, then multiply by the dividing factor and add to the next coefficient.)

These numbers become the coefficients of a polynomial one order less than the original, in this a quadratic:

x^2 -5x +125

This cannot be factored, so we use quadratic formula:

(5 +- sqrt(25 - 4*125))/2

= (5+- sqrt(-475))/2   =  (5 +-sqrt(-1)*sqrt(25)*sqrt(19))/2

= 2.5 +- 2.5i*sqrt(19)

So the fully factored polynomial is

(2.5 + 2.5i*sqrt(19)) (2.5 - 2.5i*sqrt(19)) (x + 5)