# How do you use the rational roots theorem to find all possible zeros of # P(x) = x^5 + 3x^3 + 2x - 6#?

##### 1 Answer

The "possible" rational zeros are

The only real zero is

#### Explanation:

Given:

#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.

Trying

In fact it has only non-real complex zeros.