SOLUTION: Determine the intervals you would check to see when {{{f(x)=7x^3+10x^2-11x-6<0}}}

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Question 1204559: Determine the intervals you would check to see when f%28x%29=7x%5E3%2B10x%5E2-11x-6%3C0
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(1) Observe that f(1)=0, so x=1 is one root.

(2) Use synthetic division to find the remaining polynomial when the factor (x-1) is removed.

   1 |  7  10  -11  -6
     |      7   17   6
     +-----------------
        7  17    6   0

(3) Factor the remaining polynomial, 7x%5E2%2B17x%2B6, using your favorite method; or find the other roots using the quadratic formula.

7x%5E2%2B17x%2B6=%287x%2B3%29%28x%2B2%29

The other two roots are x=-3/7 and x=-2.

The roots (smallest to largest) are now -2, -3/7, and 1; the intervals you need to check are

ANSWER: The intervals to check are (-infinity,-2), (-2,-3/7), (-3/7,1), and (1, infinity).

In fact, you don't need to check all those intervals to find where the function value is negative. You know that, with a cubic polynomial with positive leading coefficient and three distinct roots, the function value will be negative on (-infinity,-2) and (-3/7,1).