How do you find the number of roots for #f(x) = x^3 + 2x^2 - 24x# using the fundamental theorem of algebra?

1 Answer
Dec 19, 2015

You can't.

Explanation:

This theorem just tells you that a polynomial #P# such that #deg(P) = n# has at most #n# different roots, but #P# can have multiple roots. So we can say that #f# has at most 3 different roots in #CC#. Let's find its roots.

1st of all, you can factorize by #x#, so #f(x) = x(x^2 + 2x - 24)#

Before using this theorem, we need to know if P(x) = #(x^2 + 2x - 24)# has real roots. If not, then we will use the fundamental theorem of algebra.

You first calculate #Delta = b^2 - 4ac = 4 + 4*24 = 100 > 0# so it has 2 real roots. So the fundamental theorem of algebra is not of any use here.

By using the quadratic formula, we find out that the two roots of P are #-6# and #4#. So finally, #f(x) = x(x+6)(x-4)#.

I hope it helped you.