# 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)=x^4+2x^3-12x^2-40x-32#?

##### 1 Answer

#### Answer:

Possible rational zeros:

#+-1, +-2, +-4, +-8, +-16, +-32#

Descartes gives us that

Actual zeros:

#### Explanation:

Given:

#f(x) = x^4+2x^3-12x^2-40x-32#

**Rational roots theorem**

By the rational roots theorem, any *rational* zeros of

That means the the only possible rational zeros are:

#+-1, +-2, +-4, +-8, +-16, +-32#

**Descartes' Rule of Signs**

The pattern of signs of the coefficients of

The pattern of signs of coefficients of

**Bonus - Find the actual zeros**

Note that the coefficient of

We find:

#f(-2) = (-2)^4+2(-2)^3-12(-2)^2-40(-2)-32 = 16-16-48+80-32 = 0#

So

#x^4+2x^3-12x^2-40x-32 = (x+2)(x^3-12x-16)#

We find that

#(-2)^3-12(-2)-16 = -8+24-16 = 0#

So

#x^3-12x-16 = (x+2)(x^2-2x-8)#

Finally, to factor the remaining quadratic, note that

#x^2-2x-8 = (x-4)(x+2)#

So:

#f(x) = (x+2)^3(x-4)#

has zeros