How do you find the number of roots for #f(x) = 4x^3 – x^2 – 2x + 1# using the fundamental theorem of algebra?
The fundamental theorem of algebra (FTOA) tells you that a polynomial of non-zero degree has at least one Complex (possibly Real) zero.
A straightforward corollary of this - often stated as part of the FTOA - is that a polynomial in one variable of degree
To see this, note that if
In our example
For more information about the zeros, we can evaluate the discriminant of the cubic.
#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#
In this example,
#Delta = 4+128+4-432+144 = -152 < 0#
We can use Descartes' rule of signs to tell that