The better version of Integral Zero Theorem
To find the factors of any polynomial, it takes long, but it is possible. You need to find all the factors of the leading coefficient, then all the factors of the constant, then the factors are: (every constant factor/every leading coefficient factor).
Theorem
P(b/a) = 0, then it is a factor
Question
P(x) = 4x^3 + 16x^2 +9x - 9 b : {+-1.+-3.+-9} a : {+-1,+-2,+-4}
This will take a while.
P(1/1) != 0
P(-1/1) != 0
P(3/1) != 0
P(-3/1) = 0!!!!!
P(9/1) != 0
P(-9/1) != 0
P(1/-1) != 0
P(-1/-1) != 0
P(3/-1) = 0!!!!!
P(-3/-1) != 0
P(9/-1) != 0
P(-9/-1) != 0
P(1/2) = 0!!!!!
P(-1/2) != 0
P(3/2) != 0
P(-3/2) = 0!!!!!
P(9/2) != 0
P(-9/2) != 0
P(1/-2) != 0
P(-1/-2) = 0!!!!
P(3/-2) = 0!!!!!
P(-3/-2) != 0
P(9/-2) != 0
P(-9/-2) != 0
P(1/4) != 0
P(-1/4) != 0
P(3/4) != 0
P(-3/4) != 0
P(9/4) != 0
P(-9/4) != 0
P(1/-4) != 0
P(-1/-4) != 0
P(3/-4) != 0
P(-3/-4) != 0
P(9/-4) != 0
P(-9/-4) != 0
so the zeroes are at -3. -1.5 and 0.5