document.write( "Question 1193231: Given that x^2 + px +q and 3x^2 + q have a common factor (x-b) where p,q and b are non zero.Show that 3p^2 + 4q = 0
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Algebra.Com's Answer #825265 by ikleyn(52812) $\"\"$ $\"About$

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\n" ); document.write( "Given that x^2 + px + q and 3x^2 + q have a common factor (x-b) where p,q and b are non zero.
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document.write( "From the condition, we can conclude that \"b\" is the root for each given polynomial of x,\r\n" );
document.write( "due to the Remainder theorem.  So,  \r\n" );
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document.write( "    b^2 + pb + q = 0      (1)\r\n" );
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document.write( "and  \r\n" );
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document.write( "    3b^2 + q = 0.         (2)\r\n" );
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document.write( "From (2),  q = -3b^2.  Substitute it into (1)\r\n" );
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document.write( "    b^2 + pb - 3b^2 = 0,\r\n" );
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document.write( "or\r\n" );
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document.write( "    2b^2 = pb.\r\n" );
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document.write( "Since b is not equal to zero ( ! given ! ), we can divide both sides by b in the last equation. We get then\r\n" );
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document.write( "    2b = p.\r\n" );
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document.write( "It implies  \r\n" );
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document.write( "    (2b)^2 = p^2  -->  4b^2 = p^2  -->  12b^2 = 3p^2.    (3).\r\n" );
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document.write( "From the other side,  from (2) we have  \r\n" );
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document.write( "    3b^2 = -q    -->                    12b^2 = -4q.     (4)\r\n" );
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document.write( "In (3) and (4), left sides are identical, so their right sides are equal:  3p^2 = -4q.\r\n" );
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document.write( "which means  3p^2 + 4q = 0.    QED.\r\n" );
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