What is the difference between the largest and smallest possible positive roots?I am faced with the following question:What is the difference between the largest and the smallest possible positive roots of 4x5+3x3−5x2+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.

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Kalmukujobvg · Accepted Answer

Step 1Setting   f  (  x  )  =  4      x    5    +  3      x    3    −  5      x    2    +  7  x  −  12, we have that      f    ′    (  x  )  =  20      x    4    +  9      x    2    −  10  x  +  7  =  20      x    4    +  (  3  x  −  5      /    3      )    2    +  38      /    9  &amp;gt;  0Step 2Hence, 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.

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

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