document.write( "Question 1151423: 1) suppose (x^2)+(y^2)=1 is a known identity. Determine if, under such assumption, the following is an identity: 2(y^6)+(x^5)=-2(x^6)+(x^5)+6(x^4)-6(x^2)+2
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\n" ); document.write( "2) suppose (x^2)+3=x+1 is a known identity. Determine if, under such assumption, the following is an identity: 6(x^5)+8=-6x+44
\n" ); document.write( "A) NOT an identity
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\n" ); document.write( "Note: can you please give me the answer and explain how did get that answer, I have no idea how to solve this type of questions. Thanks \n" ); document.write( "

Algebra.Com's Answer #773198 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
1) suppose
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\n" ); document.write( "\n" ); document.write( "is a known identity. Determine if, under such assumption, the following is an identity: \r
\n" ); document.write( "\n" ); document.write( " $\"2y%5E6%2Bx%5E5=-2x%5E6%2Bx%5E5%2B6x%5E4-6x%5E2%2B2\"$ <--the second equation
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document.write( "We want to show that if x and y are REAL numbers that satisfy the 2nd equation,\r\n" );
document.write( "they also satisfy the 1st equation.  That is to say that there are no REAL\r\n" );
document.write( "numbers which satisfy the second equation which do not satisfy the first\r\n" );
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document.write( "Suppose x and y are REAL numbers that satisfy this equation:\r\n" );
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document.write( "Subtract x^5 from both sides\r\n" );
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document.write( "Divide through by -2\r\n" );
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document.write( "Recognize the right side as the expansion of (x²-1)³\r\n" );
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document.write( "Take cube roots of both sides:\r\n" );
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document.write( "Subtract x² from both sides\r\n" );
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document.write( "Multiply through by -1\r\n" );
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document.write( "Yes it is an identity because x and y also satisfy x² + y² = 1 \r\n" );
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\r
\n" ); document.write( "\n" ); document.write( "2) suppose
\n" ); document.write( " $\"x%5E2%2B3=x%2B1\"$
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\n" ); document.write( "the following is an identity:
\n" ); document.write( " $\"6x%5E5%2B8=-6x%2B44\"$
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document.write( "We want to prove or disprove that any REAL value of x which satisfies\r\n" );
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document.write( "and therefore this\r\n" );
document.write( "  <-- second equation\r\n" );
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document.write( "also satisfies \r\n" );
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document.write( "and therefore\r\n" );
document.write( "  <-- first equation \r\n" );
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document.write( "Let x be a REAL number that satisfies\r\n" );
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document.write( "We wonder if the first polynomial on the left of the first equation is a factor\r\n" );
document.write( "of the polynomial on the left of the second one, so we divide by long division\r\n" );
document.write( "to see:\r\n" );
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document.write( "                    x^3+ x^2- x-3\r\n" );
document.write( "x^2- x+ 2)x^5+0x^4+0x^3+0x^2+ x-6\r\n" );
document.write( "          x^5- x^4+2x^3\r\n" );
document.write( "               x^4-2x^3+0x^2\r\n" );
document.write( "               x^4- x^3+2x^2\r\n" );
document.write( "                   -x^3-2x^2+ x\r\n" );
document.write( "                   -x^3+ x^2-2x\r\n" );
document.write( "                       -3x^2+3x-6\r\n" );
document.write( "                       -3x^2+3x-6\r\n" );
document.write( "                                0 \r\n" );
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document.write( "The remainder is zero, so indeed it is a factor. Therefore\r\n" );
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document.write( " is equivalent to\r\n" );
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document.write( "Every odd degree polynomial has at least one REAL solution.\r\n" );
document.write( "Let r be a REAL solution of .\r\n" );
document.write( "Then r is a REAL solution of .\r\n" );
document.write( "However r cannot satisfy  for its discriminant\r\n" );
document.write( "is negative and it has no REAL solutions.\r\n" );
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document.write( "So  is not an identity, under the assumption\r\n" );
document.write( "that  is an identity.\r\n" );
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document.write( "Edwin

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