document.write( "Question 1204559: Determine the intervals you would check to see when $\"f%28x%29=7x%5E3%2B10x%5E2-11x-6%3C0\"$ \n" ); document.write( "

Algebra.Com's Answer #840885 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "(1) Observe that f(1)=0, so x=1 is one root.

\n" ); document.write( "(2) Use synthetic division to find the remaining polynomial when the factor (x-1) is removed.

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document.write( "   1 |  7  10  -11  -6\r\n" );
document.write( "     |      7   17   6\r\n" );
document.write( "     +-----------------\r\n" );
document.write( "        7  17    6   0

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

\n" ); document.write( " $\"7x%5E2%2B17x%2B6=%287x%2B3%29%28x%2B2%29\"$

\n" ); document.write( "The other two roots are x=-3/7 and x=-2.

\n" ); document.write( "The roots (smallest to largest) are now -2, -3/7, and 1; the intervals you need to check are

\n" ); document.write( "ANSWER: The intervals to check are (-infinity,-2), (-2,-3/7), (-3/7,1), and (1, infinity).

\n" ); document.write( "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).

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