document.write( "Question 1176146: (a) Prove that if the roots of
\n" ); document.write( "x^3 + ax^2 + bx + c = 0 form an arithmetic sequence, then 2a^3 + 27c = 9ab.\r
\n" ); document.write( "\n" ); document.write( "(b) Prove that if 2a^3 + 27c = 9ab, then the roots of
\n" ); document.write( "x^3 + ax^2 + bx + c = 0 form an arithmetic sequence. \n" ); document.write( "
Algebra.Com's Answer #802324 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"x%5E3+%2B+ax%5E2+%2B+bx+%2B+c+=+0\" \r\n" );
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document.write( "Suppose the roots (which form an arithmetic sequence) are p-d, p, and p+d.\r\n" );
document.write( "Then\r\n" );
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document.write( "The sum of the roots is -a\r\n" );
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document.write( "\"%28p-d%29+%2B+p+%2B+%28p%2Bd%29+=+-a\"\r\n" );
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document.write( "\"p+-+d+%2B+p+%2B+p+%2B+d+=+-a\"\r\n" );
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document.write( "\"3p+=+-a\"\r\n" );
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document.write( "\"a=-3p\"\r\n" );
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document.write( "The sum of the products of pairs of roots is b\r\n" );
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document.write( "\"%28p-d%29p+%2B+%28p-d%29%28p%2Bd%29+%2B+p%28p%2Bd%29+=+b\"\r\n" );
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document.write( "\"p%5E2+-+dp+%2B+p%5E2+-+d%5E2+%2B+p%5E2+%2B+pd+=+b\"\r\n" );
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document.write( "\"3p%5E2+=+b\"\r\n" );
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document.write( "The product of the roots is -c\r\n" );
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document.write( "\"%28p-d%29%28p%29%28p%2B1%29+=+-c\"\r\n" );
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document.write( "\"p%28p-d%29%28p%2Bd%29+=+-c\"\r\n" );
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document.write( "\"p%28p%5E2-d%5E2%29+=+-c\"\r\n" );
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document.write( "\"p%5E3-pd%5E2+=+-c\"\r\n" );
document.write( "\"c+=+pd%5E2-p%5E3\"\r\n" );
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document.write( "So we have solved a, b, and c in terms of p and d\r\n" );
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document.write( "Substitute in the original equation:\r\n" );
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document.write( "\"p%5E3+%2B+%28-3p%29p%5E2+%2B+%283p%5E2%29p+%2B+%28p%2Ad%5E2-p%5E3%29+=+0\"\r\n" );
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document.write( "\"p%5E3-3p%5E3%2B3p%5E3%2Bpd%5E2-p%5E3=0\"\r\n" );
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document.write( "\"pd%5E2=0\"\r\n" );
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document.write( "So either p=0 or d=0\r\n" );
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document.write( "If p=0, then the roots are -d, 0, and d\r\n" );
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document.write( "Then the sum of the roots = -d+0+d = 0 = -a, so a=0\r\n" );
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document.write( "The sum of the products of pairs of roots = (-d)(0)+(-d)(d)+(0)(d)=-d2, so b=-d2\r\n" );
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document.write( "Then the product of the root is (-d)(0)(d) = 0, so c=0\r\n" );
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document.write( "That means the original equation, in this case, was really:\r\n" );
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document.write( "\"x%5E3%2B0x%5E2-d%5E2x%2B0=0\" or\r\n" );
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document.write( "\"x%5E3-d%5E2x=0\"\r\n" );
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document.write( "So we see if \"2a%5E3+%2B+27c+=+9ab\" holds true in this case:\r\n" );
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document.write( "\"matrix%281%2C3%2C2a%5E3%2B27c%2C%22%3F=%3F%22%2C9ab%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C2%280%29%5E3%2B27%280%29%2C%22%3F=%3F%22%2C9%280%29%280+%2B-+d%29%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C0%2C%22=%22%2C0%29\"\r\n" );
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document.write( "So yes it does hold when p=0\r\n" );
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document.write( "Now we see what happens when d=0.\r\n" );
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document.write( "Then the roots are p-0, p, and p+0, or p, p, and p. \r\n" );
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document.write( "So the three roots are all equal.\r\n" );
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document.write( "The sum of the roots is 3p, so a=-3p\r\n" );
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document.write( "The sum of the products of pairs of roots = (p)(p)+(p)(p)+(p)\r\n" );
document.write( "(p)=3p2, so b=3p2\r\n" );
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document.write( "Then the product of the roots is (p)(p)(p) = p3, so c=-p3\r\n" );
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document.write( "That means the original equation, in this case, was really:\r\n" );
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document.write( "\"x%5E3%2Bax%5E2%2Bb%5E2x%2Bc=0\" or\r\n" );
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document.write( "\"x%5E3-3px%5E2%2B3p%5E2x-p%5E3=0\" which is just \"%28x-p%29%5E3=0\"\r\n" );
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document.write( "So we see if \"2a%5E3+%2B+27c+=+9ab\" holds true in this case as well:\r\n" );
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document.write( "\"matrix%281%2C3%2C2a%5E3%2B27c%2C%22%3F=%3F%22%2C9ab%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C2%28-3p%29%5E3%2B27%28-p%5E3%29%2C%22%3F=%3F%22%2C9%28-3p%29%283p%5E2%29%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C2%28-27p%5E3%29-27p%5E3%2C%22%3F=%3F%22%2C-81p%5E3%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C-54p%5E3-27p%5E3%2C%22%3F=%3F%22%2C-81p%5E3%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C-81p%5E3%2C%22%3F=%3F%22%2C-81p%5E3%29\"\r\n" );
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document.write( "So yes it does hold true in this case too.\r\n" );
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document.write( "The (a) part of the problem is proved.\r\n" );
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document.write( "If I find time I'll do (b) as well.\r\n" );
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document.write( "Edwin

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