How do you use the rational roots theorem to find all possible zeros of # P(x) = x^5 + 3x^3 + 2x - 6#?
The "possible" rational zeros are
The only real zero is
#P(x) = x^5+3x^3+2x-6#
By the rational roots theorem any rational zeros of
That means that the only possible rational zeros are:
#+-1, +-2, +-3, +-6#
In addition, note that the sum of the coefficients is zero, i.e.:
#1+3+2-6 = 0#
Hence we can deduce that
#x^5+3x^3+2x-6 = (x-1)(x^4+x^3+4x^2+4x+6)#
The remaining quartic has all positive coefficients, so any real zeros must be negative.
In fact it has only non-real complex zeros.