Finding the intervals of increase and decrease of x 4 − x 3 − 8 x 2 − x − 6 .I tried to find the derivative by the quotient rule to obtain the critical points but the formula was getting complicated, I know that D f = R ∖ { − 2 , 3 } but then what?

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Finding the intervals of increase and decrease of                     x        4            −              x        3            −      8                      x        2            −      x      −      6      .I tried to find the derivative by the quotient rule to obtain the critical points but the formula was getting complicated, I know that       D    f    =      R    ∖  {  −  2  ,  3  } but then what?

Kason Gonzales · Accepted Answer

Step 1One can simplify calculation of derivative by partitioning fraction:                    x        4            −              x        3            −      8                      x        2            −      x      −      6        =  A      x    2    +  B  x  +  C  +      D          x      −      3        +      E          x      +      2                          x        4            −              x        3            −      8                      x        2            −      x      −      6        =      x    2    +  6  +      46          5      (      x      −      3      )        −      16          5      (      x      +      2      )                  (                                    x            4                    −                      x            3                    −          8                                      x            2                    −          x          −          6                    )        ′    =  2  x  −      46          5      (      x      −      3              )        2              +      16          5      (      x      +      2              )        2            Unfortunately, for this problem one needs to have common denominator, which makes almost no sense in proposed way of calculation. Common denominator is   5  (  x  −  3      )    2    (  x  +  2      )    2   and numerator is   2  x  (  x  −  3      )    2    (  x  +  2      )    2    −  46  (  x  +  2      )    2    +  16  (  x  −  3      )    2    =  2  (      x    5    −  2      x    4    −  11      x    3    +  9      x    2    +  8  x  −  4  )Step 2Then getting answer requires solving fifth degree algebraic equation, which I suppose cannot be done in closed-form.If there is typo error in some one of signs before       x    3   or x in problem, then this way of solution leads to cubic equation.

Finding the intervals of increase and decrease of (x^4-x^3-8)/(x^2-x-6)

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