document.write( "Question 1036226: Let C be the plane curve y = f(x) defined by the cubic function f(x) = x³ - 4x² + ax + b with a, b real constants \r
\n" ); document.write( "\n" ); document.write( "1) When C is tangent to the x-axis at x=3, what are a and b?\r
\n" ); document.write( "\n" ); document.write( "2) When (1) holds, find all x such that C has points in common with the x-axis.\r
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Algebra.Com's Answer #650898 by Edwin McCravy(20055) $\"\"$ $\"About$

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document.write( "f(x) = x³ - 4x² + ax + b\r\n" );
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document.write( "tangent to the x-axis at x=3,\r\n" );
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document.write( "Therefore 3 is a double zero, (zero of even \r\n" );
document.write( "multiplicity, which can only be multiplicity 2,\r\n" );
document.write( "Therefore synthetic division will produce a \r\n" );
document.write( "remainder of 0 twice:\r\n" );
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document.write( "  3|1 -4  a     b\r\n" );
document.write( "   |   3 -3   3a-9  \r\n" );
document.write( "    1 -1 a-3  3a+b-9 = 0\r\n" );
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document.write( "  3|1 -1 a-3\r\n" );
document.write( "   |   3  6\r\n" );
document.write( "    1  2 a+3 = 0\r\n" );
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document.write( "So f(x) factors as \r\n" );
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document.write( "f(x) = (x-3)²(x+2)\r\n" );
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document.write( "So setting both those remainders = 0, gives\r\n" );
document.write( "is a system of those two equations in two unknowns:\r\n" );
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document.write( "        a+3 = 0\r\n" );
document.write( "          a = -3\r\n" );
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document.write( "      a+b-3 = 0\r\n" );
document.write( "     -3+b-3 = 0\r\n" );
document.write( "        b-6 = 0\r\n" );
document.write( "          b = 6\r\n" );
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document.write( "f(x) = x³ - 4x² + ax + b\r\n" );
document.write( "f(x) = x³ - 4x² - 3x + 18\r\n" );
document.write( " and its factorization is:\r\n" );
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document.write( "f(x) = (x-3)²(x+2)\r\n" );
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document.write( "The zeros are 3 and -2.  \r\n" );
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document.write( "The curve C and the x-as axis have points (3,0) and (-2,0)\r\n" );
document.write( "in common. \r\n" );
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document.write( "To find the area bounded by C and the x-axis is given by\r\n" );
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document.write( "                       |3\r\n" );
document.write( "   | \r\n" );
document.write( "                       |-2\r\n" );
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document.write( "   \r\n" );
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document.write( "Approximately 52.083\r\n" );
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document.write( "Edwin

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