document.write( "Question 1209255: The roots of Ax^2+Bx+1 are the same as the roots of x^2 - 3x + 5 + 4x^2 - x - 4. What is A+B? \n" ); document.write( "

Algebra.Com's Answer #847926 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The roots of Ax^2+Bx+1 are the same as the roots of x^2 - 3x + 5 + 4x^2 - x - 4. What is A+B?
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\n" ); document.write( "\n" ); document.write( "        The hidden meaning of the \" given \" part in this problem is that \r
\n" ); document.write( "\n" ); document.write( "        the set of all complex roots of one polynomial is the same \r
\n" ); document.write( "\n" ); document.write( "        as the set of all complex roots of the other polynomial.\r
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document.write( "Reduce the long polynomial to the standard form of a quadratic polynomial by combining \r\n" );
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document.write( "like terms.  You will get  5x^2 - 4x + 1.\r\n" );
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document.write( "Thus this shortened polynomial has the same roots as the polynomial Ax^2 + Bx + 1.\r\n" );
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document.write( "The fact that two polynomials of the same degree with real coefficients have the same roots\r\n" );
document.write( "means that these polynomials have proportional coefficients \r\n" );
document.write( "(one common proportionality factor for two sets of coefficients).\r\n" );
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document.write( "But these polynomials have one common value  \"1\"  of the constant terms.\r\n" );
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document.write( "It implies that co-named coefficients in these polynomials are equal:\r\n" );
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document.write( "    A = 5,  B = -4.\r\n" );
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document.write( "So, A + B = 5 + (-4) = 1.      ANSWER\r\n" );
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