document.write( "Question 1181501: Check that (2 + i) is a root of (z^4) + (2(z^3)) - (9(z^2)) - 10z + 50 = 0. What are the remaining 3 roots? \n" ); document.write( "

Algebra.Com's Answer #811414 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "Check that (2 + i) is a root of (z^4) + (2(z^3)) - (9(z^2)) - 10z + 50 = 0. What are the remaining 3 roots?
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\n" ); document.write( "\n" ); document.write( " Yes, from the first glance, it is almost impregnable fortress.\r
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document.write( "Our polynomial is with real  ( even with integer (!) )  coefficients.\r\n" );
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document.write( "Hence, if (2+i) is a root, then (2-i) is a root, too (!)\r\n" );
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document.write( "If so, then our polynomial must be divisible by  (z-(2+i))*(z-(2-i)) = ((z-2)-i)*((z-2)-i) = (z-2)^2 + 1 = z^2 - 2z +5.\r\n" );
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document.write( "Lets make long division to check if it is TRUE:\r\n" );
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document.write( "     =      with NO REMAINDER (!)\r\n" );
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document.write( "Thus we checked that the given polynomial is a multiple of the polynomial z^2 - 2z + 5.\r\n" );
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document.write( "It means that (2+i) REALLY is a root of the given polynomial (without direct calculations (!))\r\n" );
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document.write( "    +------------------------------------------------------------------------------------------------+\r\n" );
document.write( "    |    So, we know now that                                                                        |\r\n" );
document.write( "    |                                                                                                |\r\n" );
document.write( "    |        (2+i) really is the root of the given polynomial;                                       |\r\n" );
document.write( "    |        (2-i) is the other root;                                                                |\r\n" );
document.write( "    |        one factor to the given polynomial is  z^2 - 2z + 5  with the roots  (2+i) and (2-i);   |\r\n" );
document.write( "    |        the other factor is the polynomial  z^2 +6z + 10.                                       |\r\n" );
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document.write( "At this point, the last step to do is to find the roots of the quadratic polynomial  z^2 + 6z + 10.\r\n" );
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document.write( "Apply the quadratic formula to get\r\n" );
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document.write( "         =  =  =  = -3 +- i.\r\n" );
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document.write( "The problem is just solved in full.\r\n" );
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document.write( "We checked and proved that 2+i is the root.\r\n" );
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document.write( "From it, we concluded that 2-i is the root, too.\r\n" );
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document.write( "And finally, we found two remaining roots  -3+i  and  -3-i.\r\n" );
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T R I U M P H (!)

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\n" ); document.write( "\n" ); document.write( "In his post, tutor @greenestamps writes (cited)\r
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document.write( "    In the hopes of making the problem easier, we can, as the other tutor did, assume (\"hope\") that the polynomial has real coefficients.\r\n" );
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\n" ); document.write( "\n" ); document.write( "I want to HIGHLIGHT that it is not an assumption: the given polynomial REALLY has integer (hence, real) coefficients.\r
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\n" ); document.write( "\n" ); document.write( "It is not an assumption: it is a FACT.\r
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