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:
$\frac{1}{4}$
Best Answer
Setting $f(x) = 4x^5 + 3x^3 -5x^2 + 7x - 12$, we have that $$f'(x) = 20x^4 + 9x^2 - 10x + 7 = 20x^4 + (3x-5/3)^2 + 38/9 > 0$$ Hence, $f(x)$ is an increasing function with odd degree. Hence, it has only one root. Further, $f(0) = -12$, which implies that the lone root has to be positive. Hence, the difference between the largest positive and smallest positive root is $0$.