document.write( "Question 1172123: For what value(s) of k will x^2+4x+4 be a factor of: f(x)= x^4−18x^2+kx+44\r
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Algebra.Com's Answer #797026 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Notice that   = . \r\n" );
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document.write( "Therefore, if the trinomial   divides  the polynomial  f(x)=   with no remainder,\r\n" );
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document.write( "it implies that the binomial  (x+2)  divides  this polynomial   with no remainder.\r\n" );
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document.write( "In turn, due to the Remainder theorem, it means that x= -2 is the root of the polynomial f(x).\r\n" );
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document.write( "So, we substitute  x= -2 into the polynomial f(x) and equate it to zero\r\n" );
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document.write( "    f(-2) =  = -12 - 2k = 0.\r\n" );
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document.write( "It gives us a linear equation for k, which leads to \r\n" );
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document.write( "    2k = -12,\r\n" );
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document.write( "     k = -12/2 = -6.\r\n" );
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document.write( "At this point, we obtain the following result\r\n" );
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document.write( "    +-----------------------------------------------------------------+\r\n" );
document.write( "    |  if the trinomial  x^2+4x+4  divides  the polynomial            |  \r\n" );
document.write( "    |                                                                 |\r\n" );
document.write( "    |  f(x)= x^4-18x^2+kx+44  with no remainder, then k= -6.          |\r\n" );
document.write( "    +-----------------------------------------------------------------+\r\n" );
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document.write( "But the direct check shows that \r\n" );
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document.write( "    +-----------------------------------------------------------------+\r\n" );
document.write( "    |    the trinomial  x^2+4x+4  DOES NOT  divide the polynomial     |\r\n" );
document.write( "    |                                                                 |\r\n" );
document.write( "    |            f(x)= x^4-18x^2-6x+44    with no remainder           |\r\n" );
document.write( "    +-----------------------------------------------------------------+\r\n" );
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document.write( "Hence, the value of \"k\" satisfying the given condition DOES NOT EXIST.\r\n" );
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document.write( "ANSWER.  The value of \"k\" satisfying the given condition DOES NOT EXIST.\r\n" );
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