document.write( "Question 1186259: f(x) = x^3 - 2 and g(x) = x^2 - 5x. Solve gf(x) = 6. \n" ); document.write( "
Algebra.Com's Answer #817136 by ikleyn(52803)\"\" \"About 
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\n" ); document.write( "f(x) = x^3 - 2 and g(x) = x^2 - 5x. Solve gf(x) = 6.
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\n" ); document.write( "\n" ); document.write( "            I am  179%  sure that the problem in the post is written incorrectly.\r
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\n" ); document.write( "\n" ); document.write( "            The correct writing is THIS\r
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document.write( "              f(x) = x^3 - 2 and g(x) = x^2 - 5x,  solve gof(x) = 6.      (1)\r\n" );
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\n" ); document.write( "\n" ); document.write( "            So,  the left side of the equation is the  composition  of polynomials  gof,  and not their product  gf.\r
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document.write( "Introduce new variable y = x^3 - 2.\r\n" );
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document.write( "Then the given equation (1)  takes the form\r\n" );
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document.write( "    y^2 - 5y = 6\r\n" );
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document.write( "or\r\n" );
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document.write( "    y^2 - 5y - 6 = 0.\r\n" );
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document.write( "Factor left side\r\n" );
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document.write( "    (y-6)*(y+1) = 0,\r\n" );
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document.write( "which gives the roots  y= 6 and y= -1.\r\n" );
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document.write( "If  y= 6,   then  x^3 - 2 = 6,   x^3 = 6 + 2 = 8,   which implies   x = \"root%283%2C8%29\" = 2.\r\n" );
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document.write( "If  y= -1,  then  x^3 - 2 = -1,  x^3 = -1 + 2 = 1,  which implies   x = \"root%283%2C1%29\" = 1.\r\n" );
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document.write( "So, the real roots of the equation (1) are the values 1 and/or 2.\r\n" );
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document.write( "If you want to get all complex roots of equation (1), you should obtain complex roots of equations\r\n" );
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document.write( "    x^3 = 8  and  x^3 = 1.\r\n" );
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document.write( "They are  x = \"2%2Acis%282pi%2F3%29\" and  \"2%2Acis%284pi%2F3%29\"  for equation x^3 = 8,  and  x = \"cis%282pi%2F3%29\" and  \"cis%284pi%2F3%29\"  for equation x^3 = 1.\r\n" );
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document.write( "Thus the full list of the solutions to equation (1) is\r\n" );
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document.write( "     1,  \"cis%282pi%2F3%29\",  \"cis%284pi%2F3%29\",  2,  \"2%2Acis%282pi%2F3%29\"  and  \"2%2Acis%284pi%2F3%29\".\r\n" );
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