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
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: