document.write( "Question 834474: Find all real zeros:
\n" ); document.write( " $\"+f%28x%29=4x%5E4-8x%5E3-19x%5E2%2B23x-6+\"$
\n" ); document.write( "Thanks for the help! \n" ); document.write( "

Algebra.Com's Answer #503033 by KMST(5328) $\"\"$ $\"About$

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$\"+f%28x%29=red%284%29x%5E4-8x%5E3-19x%5E2%2B23x-green%286%29\"$
\n" ); document.write( "The possible rational zeros can be written as $\"n%2Fd\"$ or $\"-n%2Fd\"$ where $\"n\"$ is a factor of $\"green%286%29\"$ and $\"d\"$ is a factor of $\"red%284%29\"$ .
\n" ); document.write( "They are:
\n" ); document.write( "-6,-3,-2,-1,1,2,3,6,
\n" ); document.write( "-3/2,-3/4,-1/2,-1/4,1/4,1/2,3/4,3/2.
\n" ); document.write( "Trying the easiest ones first, we find two zeros:
\n" ); document.write( " $\"f%281%29=4-8-19%2B23-6=-6\"$
\n" ); document.write( " $\"f%28-1%29=4%2B8-19-23-6=-36\"$
\n" ); document.write( " $\"f%282%29=64-64-76%2B46-6=-36\"$
\n" ); document.write( " $\"f%28-2%29=64%2B64-76-46-6=0\"$
\n" ); document.write( " $\"f%283%29=324-216-171%2B69-6=0\"$
\n" ); document.write( "Since $\"f%28-2%29=0\"$ and $\"f%283%29=0\"$ ,
\n" ); document.write( " $\"%28x-%28-2%29%29=%28x%2B2%29\"$ and $\"%28x-3%29\"$ are factors of $\"f%28x%29\"$ .
\n" ); document.write( "So, we can divide $\"f%28x%29\"$ by $\"%28x%2B2%29\"$ and by $\"%28x-3%29\"$ one after the other,
\n" ); document.write( "or we can divide $\"f%28x%29\"$ by $\"%28x%2B2%29%28x-3%29=x%5E2-x-6\"$ .
\n" ); document.write( "Either way, the final quotient is $\"4x%5E2-4x%2B1=%282x-1%29%5E2\"$ ,
\n" ); document.write( "which has $\"x=1%2F2\"$ as a double zero.
\n" ); document.write( "So $\"f%28x%29=%28x%2B2%29%28x-3%29%282x-1%29%5E2\"$ , and the zeros of $\"f%28x%29\"$ are
\n" ); document.write( " $\"x=-2\"$ , $\"x=3\"$ , and $\"x=1%2F2\"$ . \n" ); document.write( "

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