document.write( "Question 1189239: Given that $\"+x%5E2-3x%2B2+\"$ is a factor of $\"+x%5E4+%2B+kx%5E3+-+10x%5E2+-+20x%2B24+\"$ evaluate the sum of the four roots of the equation $\"+x%5E4+%2B+kx%5E3+-+10x%5E2+-+20x%2B24+=+0+\"$ \n" ); document.write( "

Algebra.Com's Answer #820545 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Given that $\"+x%5E2-3x%2B2+\"$ is a factor of $\"+x%5E4+%2B+kx%5E3+-+10x%5E2+-+20x%2B24+\"$ evaluate
\n" ); document.write( "the sum of the four roots of the equation $\"+x%5E4+%2B+kx%5E3+-+10x%5E2+-+20x%2B24+=+0+\"$
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document.write( "Notice that x^2-3x+2 = (x-2)*(x-1).\r\n" );
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document.write( "We are given that the polynomial  x^4 + kx^3 - 10x^2 - 20x+24  is divisible by the polynomial  x^2-3x+2 .\r\n" );
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document.write( "Hence, the polynomial  x^4 + kx^3 - 10x^2 - 20x+24  is divisible by  (x-1).\r\n" );
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document.write( "It means (the Remainder theorem) that the number x= 1 is the root of the polynomial  x^4 + kx^3 - 10x^2 - 20x+24.\r\n" );
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document.write( "So, we substitute x= 1 into this polynomial, and we get this equation for \"k\"\r\n" );
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document.write( "    1^4 + k*1^3 - 10*1^2 - 20*1 + 24 = 0,\r\n" );
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document.write( "or\r\n" );
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document.write( "    1 + k - 10 - 20 + 24 = 0,\r\n" );
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document.write( "    k = 5.\r\n" );
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document.write( "Now use the Vieta's theorem: the sum of the roots of the polynomial  x^4 + kx^3 - 10x^2 - 20x + 24  is equal \r\n" );
document.write( "to the coefficient at x^3 with the opposite sign.\r\n" );
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document.write( "It gives that the sum of the roots of the polynomial  x^4 + kx^3 - 10x^2 - 20x+24   is equal to -k, i.e. -5.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "To tutor @MathTherapy: thanks for checking my post !\r
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