document.write( "Question 825866: Find all real solutions of the polynomial equation x^4-3x^3+6x-4=0 by using the Rational Zero Theorem and Synthetic division.
\n" ); document.write( "(The sollution to this problem is +1, +2, sqrt(2) and -(sqrt(2)) )\r
\n" ); document.write( "\n" ); document.write( "Possible rational zeros: +-1, +-2, +-4
\n" ); document.write( "So I did Synthetic division, and +1 worked out. But when I tried +2 with the quotient of the first division, the remainder was 4 which I hadn't expected because according to the answer sheet +2 is a zero.
\n" ); document.write( "Please help. \n" ); document.write( "

Algebra.Com's Answer #497716 by josgarithmetic(39620) $\"\"$ $\"About$

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Both polynomial division and synthetic division require accounting for each variable placement value. The dividend should be $\"x%5E4-3x%5E3%2B0%2Ax%5E2-4\"$ , in case you missed one of the terms.\r
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\n" ); document.write( "\n" ); document.write( "I will give some results here, but because of slow task of writing on the site in text, my work is done separately on paper.\r
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\n" ); document.write( "\n" ); document.write( "The first goal is find all of the rational roots. Later, the irrational ones are found using general solution to quadratic equation.\r
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\n" ); document.write( "\n" ); document.write( "Test for possible roots, -1,-2,-4,1,2,4; for each you find, your unfound roots become much fewer.\r
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\n" ); document.write( "\n" ); document.write( "(Synthetic Division used):
\n" ); document.write( "The check first for root of +1 gives remainder 0, resulting $\"1%2Ax%5E3-2x%5E2-2x%2B4\"$ ;
\n" ); document.write( "A check for root of +2 gives remainder 0, resulting $\"1%2Ax%5E2%2B0%2Ax-2\"$ ;
\n" ); document.write( "This last expression is for the factor $\"x%5E2-2=0\"$ , which has simply the solutions $\"sqrt%282%29\"$ and $\"-sqrt%282%29\"$ \n" ); document.write( "

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