# 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)=-2x^3+19x^2-49x+20#?

##### 1 Answer

#### Answer:

See explanation...

#### Explanation:

Given:

#f(x) = -2x^3+19x^2-49x+20#

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

That means that the only possible *rational* zeros are:

#+-1/2, +-1, +-2, +-5/2, +-4, +-5, +-10, +-20#

The pattern of signs of the coefficients of

The pattern of signs of the coefficients of

Putting these together, we can deduce that the only possible *rational* zeros of

#1/2, 1, 2, 5/2, 4, 5, 10, 20#

Trying each in turn, we find:

#f(1/2) = -2(color(blue)(1/8))+19(color(blue)(1/4))-49(color(blue)(1/2))+20#

#color(white)(f(1/2)) = -1/4+19/4-98/4+80/4 = 0#

So

#-2x^3+19x^2-49x+20 = (2x-1)(-x^2+9x-20)#

#color(white)(-2x^3+19x^2-49x+20) = -(2x-1)(x-4)(x-5)#

So the three zeros are: