How do you find the number of roots for #0=x^3+450x^2-1350# using the fundamental theorem of algebra?
This polynomial equation is of degree
Actually all three roots are Real and distinct.
The fundamental theorem of algebra tells you that a polynomial equation in one variable of degree
A simple corollary of this is that a polynomial equation in one variable of degree
Here's a sketch of the proof of the corollary:
If a polynomial
Note that the Complex roots referred to in all of this may be Real or non-Real.
Beyond this, since this polynomial has Real coefficients, any non-Real Complex roots will occur in conjugate pairs. So we can deduce that our polynomial will have at least one Real root.
It so happens that all of these
To show that all of the roots are Real, let
#f(-450) = -450^3+450^3-1350 = -1350 < 0#
#f(-2) = -8+450*4-1350 = -8+1800-1350 = 442 > 0#
#f(0) = 0 + 0 - 1350 = -1350 < 0#
#f(2) = 8+450*4-1350 = 8+1800-1350 = 458 > 0#