How do you find the number of roots for #f(x) = x^4 + 5x^3 +10x^2 + 20x + 24# using the fundamental theorem of algebra?
The FTOA tells us that there are four zeros counting multiplicity.
Other methods help us find them:
The fundamental theorem of algebra tells you that any non-constant polynomial in one variable with Real or Complex coefficients has a zero in the field of Complex numbers,
A corollary of this, often stated as part of the FTOA, is that a polynomial with Complex coefficients of degree
In our example,
What else can we find out about these zeros?
First note that all of the coefficients of
If we reverse the signs on the terms of odd degree, then the pattern of signs of coefficients is:
Using the rational root theorem, the only possible rational zeros are expressible in the form
That means that the only possible rational zeros are:
#-1, -2, -3, -4, -6, -8, -12, -24#
Trying each of these in turn, we find:
#f(-2) = 16-40+40-40+24 = 0#
#x^4+5x^3+10x^2+20x+24 = (x+2)(x^3+3x^2+4x+12)#
The remaining cubic factor factors by grouping as follows:
#x^3+3x^2+4x+12 = (x^3+3x^2)+(4x+12)#
So the zeros are: