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
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