The product of the roots of a cubic polynomial is 14 and the sum of the roots is 4. If the roots are all different integers, what is the polynomial?

Question

sum and product zeros

Answer 1

Let x, y, and z be the real roots, since we have a cubic polynomial that has at most 3 real roots.

 

xyz = 14                          eq1

x + y + z = 4                   eq2

 

The factors of 14 are 1, 2, 7, and 14.

 

 

What if we try this combination,

 

(-1)(-2)(7) = 14

 

-1 + (-2) + 7 = 4

 

 

This works out!

 

 

The roots are then  -2, -1, and 7.

 

Your polynomial is then

 

f(x) = (x + 1)(x + 2)(x - 7)