document.write( "Question 330385: (b) Solve for x, the equation 12x3 + 8x2 - x - 1 = 0. (All solutions are rational and between ą1.) \n" ); document.write( "

Algebra.Com's Answer #236929 by galactus(183) $\"\"$ $\"About$

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$\"12x%5E3%2B8x%5E2-x-1=0\"$ \r
\n" ); document.write( "\n" ); document.write( "Let's try breaking it up, grouping and factoring. One could use the Rational Root Theorem as well.\r
\n" ); document.write( "\n" ); document.write( "Rewrite:\r
\n" ); document.write( "\n" ); document.write( " $\"12x%5E3%2B12x%5E2-4x%5E2-4x%2B3x-1=0\"$ \r
\n" ); document.write( "\n" ); document.write( "Group:\r
\n" ); document.write( "\n" ); document.write( " $\"12x%5E3%2B12x%5E2%2B3x-4x%5E2-4x-1\"$ \r
\n" ); document.write( "\n" ); document.write( "Factor out 3x from the first 3 terms and -1 from the last 3 terms: \r
\n" ); document.write( "\n" ); document.write( " $\"3x%284x%5E2%2B4x%2B1%29-%284x%5E2%2B4x%2B1%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%283x-1%29%284x%5E2%2B4x%2B1%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%283x-1%29%282x%2B1%29%5E2=0\"$ \r
\n" ); document.write( "\n" ); document.write( "What makes 3x-1=0?. x=1/3\r
\n" ); document.write( "\n" ); document.write( "What makes 2x+1=0?. x=-1/2\r
\n" ); document.write( "\n" ); document.write( "Since x=-1/2 has multiplicity 2, we have three roots as required of the cubic.\r
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