What is the difference between the largest and smallest possible positive roots?

Question

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: $\frac{1}{4}$

Answer 1

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 $0$.

Answer 2

1. Restatement of Question to Make it Easier to Reference In the Solution

What is the difference between the largest and the smallest possible positive roots of $4x^5 + 3x^3 -5x^2 + 7x - 12$?

The corresponding Equation for this question is:

$$f\left(x\right)=4x^5 + 3x^3 -5x^2 + 7x - 12 =0 \tag{Eq. 1.1}$$

2. Solution Steps

Apply Descartes's Sign Rule (as quoted):

1. First, determine how many maximum positive roots: There are three sign changes going from lowest power to highest power*. So this implies a maximum of three positive roots. Or a minimum of one positive root.
2. Second, determine how many maximum negative roots. First flip the signs of the odd power coefficients resulting in: $$g\left(x\right)=f\left(-x\right)= -4x^5 - 3x^3 -5x^2 - 7x - 12 =0 \tag{Eq. 2.1} $$ There are zero sign changes so there are zero possible negative roots. It can be seen that when $x>0$ in Equation 2.1, that each of the terms contributes negatively, starting with $-12$ when $x=0$. When $x>0$ then $g(x)$ becomes even more so negative, not crossing the $y=g\left(x\right)$ line any further.

This is a fifth degree polynomial $f\left(x\right)$. If there are three positive roots, then there are two complex roots. If there is one positive root, then there are four complex roots.

The question seems to imply that there are two different positive roots, since it is asking for a subtraction of these roots.

However, the following quoted Desmos plot seems to indicate one positive root:

Therefore there may be an issue with the question but that needs to be investigated further to make a firm determination.

3. Conclusions of the Graph

Zooming in on the previous graph shows that there is only one root that occurs at $x \approx 1.1$. The question referenced two roots, the largest root and the smallest root. I consider this an issue with the question, as there is only one real root and not more. Of course the difference between the one root and itself is zero as the quoted Desmos online Calculator plot shows below:

4. Supporting Derivative Calculations

Consider:

$$f\left(x\right)=4x^5 + 3x^3 -5x^2 + 7x - 12 =0 \tag{Eq. 1.1}$$

When $x=0$ then $f=-12$. Also, the slope is positive.

The derivative $\frac1{d\,x}f(x)$ is zero at a turning point or local maximum:

$$\frac1{d\,x}f\left(x\right)=20\,x^4 + 9\,x^2 -10\,x + 7 =0 \tag{Eq. 3.2}$$

The quoted Desmos 2-Dimensional plot follows of $f'(x,y)$:

From Equation 3.2, a zero might occur for $\frac1{d\,x}f\left(x,y\right)$ in the region where $-10\,x+7<0$. To evaluate this region, set $-10\,x=-7$, or $x=0.7$; for $x<0.7$, $-10\,x + 7 > 0$, indicating no maximums in that region.

At $x=0.7$, then $\left.\frac1{d\,x}f\left(x\right)\right|_{x=0.7}=20\,\left(0.7\right)^4 + 9\,\left(0.7\right)^2 -10\,\left(0.7 \right) + 7 = 9.212 $ and it is ever-increasing.

$$\boxed{\text{So there are }\textbf{not more than one real roots }\text{ for } f\left(x\right)=4x^5 + 3x^3 -5x^2 + 7x - 12 =0}$$

And of course if there is only one real root, that root subtracted by itself is zero. But the problem statement seemed to indicate more than one real root, and that is the difficulty to the solution of the problem statement as given.