document.write( "Question 1186415: f(x) = x^3 - 2 and g(x) = x^2 - 5x. Solve gf(x) = 6 \n" ); document.write( "

Algebra.Com's Answer #817321 by ikleyn(52754) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "The polynomial equation   gf(x) = 6,   written exactly as presented in this post, has the form\r
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\n" ); document.write( "\n" ); document.write( "        (x^2-5x)*(x^3-2) = 6\r
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\n" ); document.write( "\n" ); document.write( "which is the same as \r
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\n" ); document.write( "\n" ); document.write( "        x^5 - 5x^3 - 2x^2 + 4 = 0         (after FOIL and combining like terms).\r
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\n" ); document.write( "\n" ); document.write( "As an authoritative online calculator \r
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\n" ); document.write( "\n" ); document.write( "https://www.mathportal.org/calculators/solving-equations/polynomial-equation-solver.php \r
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\n" ); document.write( "\n" ); document.write( "says, this equation HAS NO rational roots that can be found using Rational Root Theorem.\r
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\n" ); document.write( "\n" ); document.write( "They only can be found using numerical methods.\r
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\n" ); document.write( "\n" ); document.write( "Approximate values are\r
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\n" ); document.write( "\n" ); document.write( "        x1 = 0.84379\r
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\n" ); document.write( "\n" ); document.write( "        x2 = −2.1159\r
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\n" ); document.write( "\n" ); document.write( "        x3 = 2.35435\r
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\n" ); document.write( "\n" ); document.write( "        x4 = −0.54112 + 0.81167*i\r
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\n" ); document.write( "\n" ); document.write( "        x5 = −0.54112 − 0.81167*i\r
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\n" ); document.write( "\n" ); document.write( "From the other side, if to suppose that the correct form of the given equation is different, namely, is \r
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\n" ); document.write( "\n" ); document.write( "        gof(x) = 0         (1)    (composition of polynomial functions instead of their product),\r
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\n" ); document.write( "\n" ); document.write( "then the EXACT solution can be obtained ANALYTICALLY.\r
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\n" ); document.write( "\n" ); document.write( "This analytic solution is presented in other my post at this forum, where I solved this problem couples of days ago. See the link\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/playground/test.faq.question.1186259.html\r
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\n" ); document.write( "\n" ); document.write( "For your convenience, I copy/paste this solution here AGAIN.\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 =  = 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 =  = 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 =  and    for equation x^3 = 8,  and  x =  and    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,  ,  ,  2,    and  .\r\n" );
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