document.write( "Question 733800: Find a polynomial f(x) of degree 4 that has the indicated zeros and satisfies the given condition: -1, 2, 3i; and f(-2)=10\r
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Algebra.Com's Answer #448615 by Edwin McCravy(20064) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( " -1, 2, 3i; and f(-2)=10\r\n" );
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document.write( "Since 3i is a solution, its conjugate -3i is also a solution.\r\n" );
document.write( "[Note that since 3i is really 0+3i, its conjugate is 0-3i, which\r\n" );
document.write( "is -3i]\r\n" );
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document.write( "So we start with\r\n" );
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document.write( " x = -1;   x = 2,   x = 3i,  x = -3i\r\n" );
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document.write( "Get 0 on the right side of each:\r\n" );
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document.write( "  \r\n" );
document.write( "x+1 = 0;   x-2 = 0  x-3i = 0,  x+3i = 0\r\n" );
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document.write( "Myltiply all the left and right sides together.  The right side\r\n" );
document.write( "will just be 0:\r\n" );
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document.write( "   (x+1)(x-2)(x-3i)(x+3i) = 0\r\n" );
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document.write( "Multiply both sides by a constant k, and the left\r\n" );
document.write( "side will be f(x) if we have the right value for k\r\n" );
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document.write( "f(x) = k(x+1)(x-2)(x-3i)(x+3i) \r\n" );
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document.write( "Multiply the first two and the last two factors  \r\n" );
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document.write( "f(x) = k(x²-2x+1x-2)(x²+3ix-3ix-9i²) \r\n" );
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document.write( "f(x) = k(x²-x-2)(x²-9i²) \r\n" );
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document.write( "Now since i² = -1, -9i² = -9(-1) = +9\r\n" );
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document.write( "f(x) = k(x²-x-2)(x²+9) \r\n" );
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document.write( "Multiply those two parentheses together:\r\n" );
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document.write( "f(x) = k(x⁴+9x²-x³-2x²-9x-18) \r\n" );
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document.write( "Collect like terms:\r\n" );
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document.write( "f(x) = k(x⁴-x³+7x²-9x-18)\r\n" );
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document.write( "Now we can find k because we are given f(-2)=10\r\n" );
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document.write( "We substitute x=-2\r\n" );
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document.write( "f(-2) = k((-2)⁴-(-2)³+7(-2)²-9(-2)-18)\r\n" );
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document.write( "f(-2) = k(16-(-8)+7(4)+18-18)\r\n" );
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document.write( "f(-2) = k(16+8+28)\r\n" );
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document.write( "f(-2) = k(52)\r\n" );
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document.write( "We substitute 10 for f(-2)\r\n" );
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document.write( "  10 = k(52)\r\n" );
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document.write( " = k       \r\n" );
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document.write( " = k\r\n" );
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document.write( "Substitute for k in:\r\n" );
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document.write( "f(x) = k(x⁴-x³+7x²-9x-18)\r\n" );
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document.write( "f(x) = (x⁴-x³+7x²-9x-18)\r\n" );
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document.write( "f(x) = x⁴ - x³ + x² - x - \r\n" );
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document.write( "Edwin

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