I am faced with the following question:
What is the difference between the largest and the smallest possible positive roots of $4x^5 + 3x^3 -5x^2 + 7x - 12$?
Now, my first attempt was to try substituting arbirtrary values to find one root and then long division to find the others. However, no integer (or fractional) value seemed to satisfy this.
Is the another way to approach this problem, or am I just making a simple arithmetic mistake?
Any help will be appreciated.
EDIT
Here is the official solution:
The possible roots of a polynomial can be determined by finding all combinations of quotients with the numerator being a factor of the constant and the denominator being a factor of the leading coefficient. However, we don’t need to consider all factors, just the largest and smallest. The largest possibility will come from the largest numerator and smallest denominator and the smallest will come from the smallest numerator and largest denominator. The largest will always be the number itself and the smallest will always be 1.
The largest possible root: $\frac{12}{1}$ The smallest possible root: $\frac{1}{4}$