Question 1204559
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(1) Observe that f(1)=0, so x=1 is one root.<br>
(2) Use synthetic division to find the remaining polynomial when the factor (x-1) is removed.<br><pre>

   1 |  7  10  -11  -6
     |      7   17   6
     +-----------------
        7  17    6   0</pre>
(3) Factor the remaining polynomial, {{{7x^2+17x+6}}}, using your favorite method; or find the other roots using the quadratic formula.<br>
{{{7x^2+17x+6=(7x+3)(x+2)}}}<br>
The other two roots are x=-3/7 and x=-2.<br>
The roots (smallest to largest) are now -2, -3/7, and 1; the intervals you need to check are<br>
ANSWER: The intervals to check are (-infinity,-2), (-2,-3/7), (-3/7,1), and (1, infinity).<br>
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).<br>