Roots of cubic functions

Question

Consider a polynomial

p(x) = x

3 + ax2 + bx + c

What sufficient conditions for a,b,c will hold in order to find the roots

of p(x) just by factorization only?

(a) a

b

=

1

c

(b) a

c

= b

(c) b

c

= a

(d) a

c

=

1

b

Answer 1

First you must specify what the characteristics of a,b, and c are.

I think the question presumes a, b and c are real integers.

Since the coefficient of the cubic term is 1, there can be no real fractional roots.

To solve by factoring, you want the roots to be real integers.

Then any root must be a factor of c.

Beyond that I am not sure of any conditions on a and b,

However, if you convert to the reduced cubic (i.e. no squared term), which can always be done,

then Cardan's formula will tell you when the roots are real.