document.write( "Question 1172365: Sam is working on some polynomial factorizations in the form of x^2 + px + q , where p and q are nonzero integers and \"a\" and \"b\" are integer numbers such that x^2+px+q = (x+a)(x+b). His work is as follows:
\n" ); document.write( "x^2 −2x−3=(x−3)(x+1)
\n" ); document.write( "x^2 +5x+6=(x+2)(x+3)
\n" ); document.write( "x^2 −7x+10=(x−2)(x−5)
\n" ); document.write( "x^2 +6x+8=(x+2)(x+4)
\n" ); document.write( "x^2 −8x+12=(x−2)(x−6)
\n" ); document.write( "x^2 +9x+18=(x+3)(x+6)
\n" ); document.write( "He concludes that if p and q are coprime, then the factors a and b are also coprime. If p and q are not coprime, then the factors a and b are not coprime, either.
\n" ); document.write( "Is his conclusion correct? Explain please. \n" ); document.write( "
Algebra.Com's Answer #797386 by ikleyn(52788)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "So,  we have two integer numbers,  \"a\"  and  \"b\",   on one hand side,  and\r
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\n" ); document.write( "\n" ); document.write( "         we have two other integer p and q,  on the other hand side,  such that \r
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\n" ); document.write( "\n" ); document.write( "                 p = -(a+b)   and   q = ab.\r
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\n" ); document.write( "\n" ); document.write( "(1)    First question is:   is it true that  IF  p  and  q  are coprime,  THEN  \"a\"  and  \"b\"  are coprime ?\r
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document.write( "     Let's prove it by CONTRADICTION.\r\n" );
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document.write( "         Let assume that \"a\" and \"b\" are not coprime.\r\n" );
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document.write( "         It means that there is such a prime number w such that \"a\" and \"b\" are multiple of w.\r\n" );
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document.write( "         Then it is OBVIOUS that w is the common divisor for (a+b) and ab.\r\n" );
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document.write( "         Hence, w is the common divisor for p and q.\r\n" );
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document.write( "         It proves, by CONTRADICTION, that if p and q are coprime, then \"a\" and \"b\" are coprime, too.\r\n" );
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document.write( "     So, the first statement is proved.\r\n" );
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\n" ); document.write( "\n" ); document.write( "(2)    Second statement is:   IF  p  and  q  are not coprime,  THEN  \"a\"  and  \"b\"  are not coprime,  too.\r
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document.write( "    Here the DIRECT proof works.\r\n" );
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document.write( "        If p and q are not coprime, then (a+b) and (ab)  are not coprime.\r\n" );
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document.write( "        It means that (a+b) and (ab) have common prime divisor w.\r\n" );
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document.write( "        Since w divides the product (ab), it divides at least one of the numbers \"a\" or \"b\".\r\n" );
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document.write( "            If it eventually divides both \"a\" and \"b\", then there is NOTHING to prove . . . \r\n" );
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document.write( "        So, let assume that w divides \"a\". \r\n" );
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document.write( "        Then w divides \"a\" and (a+b),  at the same time.\r\n" );
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document.write( "        It just implies that w divides b, too   (which is OBVIOUS).\r\n" );
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document.write( "        So, we proved that if p and q are not coprime, then \"a\" and \"b\" are not coprime, too.\r\n" );
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document.write( "    The second statement is proved.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problem is just solved:   both statements are proved.\r
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\n" ); document.write( "\n" ); document.write( "The post-solution note:   do not call the numbers  \"a\"  and  \"b\"  as   factors  -  in given context,  they are not.\r
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