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.


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: $\frac{1}{4}$

  • 1
    $\begingroup$ "possible" roots ? $\endgroup$ – Yves Daoust Apr 10 '15 at 17:06
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    $\begingroup$ Have you tried sketching the function to see what is gong on? $\endgroup$ – Mark Bennet Apr 10 '15 at 17:17
  • $\begingroup$ @MarkBennet yes I have tried that, but to no avail. Is there something I should be seeing if I do that? $\endgroup$ – Varun Iyer Apr 10 '15 at 17:19
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    $\begingroup$ list down all the positive factors of constant term and leading coefficient : $$12 : 1,2,3,4,6,12\\4 : 1,2,3,4$$ $\endgroup$ – AgentS Apr 10 '15 at 17:31
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    $\begingroup$ The question is misleading and is not about actual roots of the polynomial at all. Rather it is referring to the application of the rational root theorem to determine the range within which a positive root must lie. Do you know the rational root theorem, and does it form part of the context in which the question is set? If so, this is important information which is missing from your question and which should be added. $\endgroup$ – Mark Bennet Apr 10 '15 at 17:32

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 $0$.

  • $\begingroup$ The answer provided was not $0$, so I'm sorry to tell to you that your answer is incorrect. $\endgroup$ – Varun Iyer Apr 10 '15 at 17:18
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    $\begingroup$ @VarunIyer Clearly what you say is incorrect. See WA with the function plotted. There is only one positive root. $\endgroup$ – Leg Apr 10 '15 at 17:20
  • $\begingroup$ what do you mean. I already understand that there is only one positive roots. There are also 4 COMPLEX roots. Those can also be considered yes? So how can the answer be $0$? $\endgroup$ – Varun Iyer Apr 10 '15 at 17:24
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    $\begingroup$ @VarunIyer There is no positive or negative complex number. Neither are large and small complex numbers. $\endgroup$ – Leg Apr 10 '15 at 17:25
  • $\begingroup$ I have posted the official solution. $\endgroup$ – Varun Iyer Apr 10 '15 at 17:26

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