# How do you use the Rational Zeros theorem to make a list of all possible rational zeros, and use the Descarte's rule of signs to list the possible positive/negative zeros of #f(x)=36x^4-12x^3-11x^2+2x+1#?

##### 1 Answer

#### Explanation:

Given:

#f(x) = 36x^4-12x^3-11x^2+2x+1#

By the rational zeros theorem, any rational zeros of

That means that the only possible rational zeros are:

#+-1/36, +-1/18, +-1/12, +-1/9, +-1/6, +-1/4, +-1/3, +-1/2, +-1#

Note that the pattern of signs of the coefficients of

The pattern of signs of the coefficients of

Note that if we reverse the order of the coefficients, we get a polynomial whose zeros are the reciprocals of the zeros of

#g(x) = x^4+2x^3-11x^2-12x+36#

This has possible rational zeros:

#+-1, +-2, +-3, +-4, +-6, +-9, +-12, +-18, +-36#

(I prefer doing arithmetic with whole numbers)

We find:

#g(2) = (2)^4+2(2)^3-11(2^2)-12(2)+36 = 16+16-44-24+36 =0#

So

#x^4+2x^3-11x^2-12x+36 = (x-2)(x^3+4x^2-3x-18)#

We find that

#(2)^3+4(2)^2-3(2)-18 = 8+16-6-18 = 0#

So

#x^3+4x^2-3x-18 = (x-2)(x^2+6x+9)#

We can recognise the remaining quadratic factor as a perfect square trinomial:

#x^2+6x+9 = (x+3)^2#

So the last two zeros of

So

and