document.write( "Question 1122638: The slope of the tangent line to the graph of a cubic polynomial is -24 at two different points: (-2,79) and (1,-20). Determine the equation of the cubic polynomial. \n" ); document.write( "

Algebra.Com's Answer #738834 by ikleyn(52776) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " The solution by @josmicely is TOTALLY and ABSOLUTELY wrong.\r
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\n" ); document.write( "\n" ); document.write( " Below find the correct solution.\r
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\n" ); document.write( "\n" ); document.write( "We are going to find the numerical values of four coefficients a, b, c and d of the polynomial of the third degree y = ax^3 + bx^2 + cx + d.\r
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document.write( "Equation 1.\r\n" );
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document.write( "    I will derive this equation from the condition that the point (-2,79) lies on the graph, i.e. satisfies the equation\r\n" );
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document.write( "    a*(-2)^3 + b*(-2)^2  + c*(-2) + d = 79,   or\r\n" );
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document.write( "    -8a + 4b -2c + d = 79.          (1)\r\n" );
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document.write( "Equation 2.\r\n" );
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document.write( "    I will derive this equation from the condition that the point (1,-20) lies on the graph, i.e. satisfies the equation\r\n" );
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document.write( "    a*1^3 + b*1^2  + c*1 + d = -20,   or\r\n" );
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document.write( "    a + b + c + d = -20.            (2)\r\n" );
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document.write( "Equation 3.\r\n" );
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document.write( "    I will derive this equation from the condition that the slope at the point (-2,79) is -24.\r\n" );
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document.write( "    Since the first derivative is y'(x) = 3ax^2 + 2bx + c,  the equation for the slope is\r\n" );
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document.write( "    3a*(-2)^2 + 2b*(-2)  + c = -24,   or\r\n" );
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document.write( "    12a - 4b + c = -24.             (3)\r\n" );
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document.write( "Equation 4.\r\n" );
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document.write( "    I will derive this equation from the condition that the slope at the point (1,-20) is -24.\r\n" );
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document.write( "    Similarly to equation 3 case,  the equation for the slope is\r\n" );
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document.write( "    3a*1^2 + 2b*1  + c = -24,   or\r\n" );
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document.write( "    3a + 2b + c = -24.               (4)\r\n" );
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document.write( "So, your system of equations is  \r\n" );
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document.write( "     -8a + 4b -2c + d =  79         (1)\r\n" );
document.write( "       a +  b + c + d = -20         (2)\r\n" );
document.write( "     12a - 4b + c     = -24         (3)\r\n" );
document.write( "      3a + 2b + c     = -24.        (4)\r\n" );
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document.write( "Next, use some technique or technology to solve it.\r\n" );
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document.write( "Probably, your hand calculator may help.\r\n" );
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document.write( "I use an online free of charge matrix equation solver\r\n" );
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document.write( "https://matrix.reshish.com/gauss-jordanElimination.php\r\n" );
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document.write( "I recommend you to get familiar with it.\r\n" );
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document.write( "To complete your assignment, input the augmented matrix into the solver and press the \"Solve\" button.\r\n" );
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