document.write( "Question 825414: Find all the zeros of the polynomial function.\r
\n" ); document.write( "\n" ); document.write( "f(x) = $\"+x%5E3-4x%5E2-11x%2B2\"$ \n" ); document.write( "

Algebra.Com's Answer #497361 by jsmallt9(3758) $\"\"$ $\"About$

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The possible rational zeros are +1/1 and +2/1 (which simplify to +1 and +2). Trying these we find that only -2 is actually a zero:
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document.write( "-2  |   1   -4   -11   2\r\n" );
document.write( "-----       -2    12  -2\r\n" );
document.write( "       ------------------\r\n" );
document.write( "        1   -6     1   0\r\n" );
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The zero in the lower right corner is the remainder. (It is also f(-2)!) Since it is zero then what we divided by (x-(-2)) [or (x+2)] divides evenly. So (x+2) is a factor of f(x). Not only that but the rest of the bottom row tells us what the other factor is. The \"1 -6 1\" translates into $\"x%5E2-6x%2B1\"$ . So now
\n" ); document.write( " $\"f%28x%29+=+x%5E3-4x%5E2-11x%2B2+=+%28x%2B2%29%28x%5E2-6x%2B1%29\"$
\n" ); document.write( "The other zeros of f(x) will come from the second factor. Since it is a quadratic that won't factor, we can use the quadratic formula:
\n" ); document.write( " $\"x+=+%28-%28-6%29%2B-sqrt%28%28-6%29%5E2-4%281%29%281%29%29%29%2F2%281%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"x+=+%286%2B-sqrt%2836-4%281%29%281%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%286%2B-sqrt%2836-4%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%286%2B-sqrt%2832%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%286%2B-sqrt%2816%2A2%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%286%2B-sqrt%2816%29%2Asqrt%282%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%286%2B-+4%2Asqrt%282%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%282%283%2B-+2%2Asqrt%282%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28cross%282%29%283%2B-+2%2Asqrt%282%29%29%29%2Fcross%282%29\"$
\n" ); document.write( " $\"x+=+3%2B-+2%2Asqrt%282%29\"$
\n" ); document.write( "which is short for $\"x+=+3%2B2%2Asqrt%282%29\"$ or $\"x+=+3-2%2Asqrt%282%29\"$

\n" ); document.write( "So the three zeros for f(x) are: -2, $\"3%2B2%2Asqrt%282%29\"$ and $\"3-2%2Asqrt%282%29\"$ \n" ); document.write( "

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