document.write( "Question 662478: Iam trying to find all the zeros for
\n" ); document.write( "x5+6x4-x3-6x2-20x-120 found -6 \n" ); document.write( "

Algebra.Com's Answer #412286 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Having found -6 is a good start, and if you continued looking for rational roots from there, you were doomed to failure; there aren't any. The other four zeros consist of a pair of irrationals and a conjugate pair of complex numbers. The thing is, after having found the factor

, you were staring at a 4th degree polynomial that we now know has no rational factors. Woe is me! What to do now?\r
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\n" ); document.write( "\n" ); document.write( "Fortunately, this particular 4th degree polynomial is so configured that you can actually treat it like a quadratic. Read on.\r
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\n" ); document.write( "\n" ); document.write( "First let's do the synthetic division with -6:\r
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document.write( "-6  |    1    6   -1   -6  -20 -120\r\n" );
document.write( "             -6    0    6    0  120\r\n" );
document.write( "         1    0   -1    0   20    0\r\n" );
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\n" ); document.write( "\n" ); document.write( "From this we can determine that

and

are factors of

, and that -6 is indeed a real rational zero of the 5th degree polynomial.\r
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\n" ); document.write( "\n" ); document.write( "But what do we do with the 4th degree factor? We use a substitution trick. Let

and substitute into

:\r
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\r
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\n" ); document.write( "\n" ); document.write( "Et voilà! We have a quadratic that factors tidily:\r
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\r
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\n" ); document.write( "\n" ); document.write( "So the zeros are\r
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\r
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\n" ); document.write( "\n" ); document.write( "But wait! We aren't done. The problem doesn't want values of

that make the original polynomial equal to zero, it wants values of

. So reverse the substitution, thus:\r
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\r
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\n" ); document.write( "\n" ); document.write( "hence\r
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\r
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\n" ); document.write( "\n" ); document.write( "And we are done. We started with a 5th degree polynomial and found 5 zeros satisfying the Fundamental Theorem of Algebra. Time to sit around in a circle, hold hands, and sing \"Kumbaya.\"\r
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\n" ); document.write( "\n" ); document.write( "John
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\n" ); document.write( "My calculator said it, I believe it, that settles it
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$\"The$

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