document.write( "Question 438947: Hi, can someone please help me find all the zeros in order using synthetic division and quadratic formula for equation:
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Algebra.Com's Answer #303452 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
Hi, can someone please help me find all the zeros in order using synthetic division and quadratic formula for equation:
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document.write( "P(x)= 2x⁹ - x⁸ - 16x⁷ + 17x⁶ - 6x⁵ - 5x⁴+ 68x³ - 77x² + 18\r\n" );
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document.write( "Try 1 as a zero\r\n" );
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document.write( "     1|2   -1   -16   17   -6   -5   68   -77    0   18\r\n" );
document.write( "      |     2     1  -15    2   -4   -9    59  -18  -18 \r\n" );
document.write( "       2    1   -15    2   -4   -9   59   -18  -18    0\r\n" );
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document.write( "That factors P(x) as\r\n" );
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document.write( "P(x) = (x - 1)(2x⁸ + x⁷ - 15x⁶ + 2x⁵ - 4x⁴- 9x³ - 59x² - 18x - 18)\r\n" );
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document.write( "Try 1 as a zero again\r\n" );
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document.write( "     1|2    1   -15    2   -4   -9   59   -18  -18\r\n" );
document.write( "      |     2     3  -12  -10  -14  -23    36   18  \r\n" );
document.write( "       2    3   -12  -10  -14  -23   36    18    0\r\n" );
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document.write( "That factors P(x) as\r\n" );
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document.write( "P(x) = (x - 1)(x - 1)(2x⁷ + 3x⁶ - 12x⁵ - 10x⁴- 14x³ - 23x² + 36x + 18)\r\n" );
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document.write( "Try 1 as a zero again\r\n" );
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document.write( "     1|2    3   -12  -10  -14  -23   36   18 \r\n" );
document.write( "      |     2     5   -7  -17  -31  -54  -18 \r\n" );
document.write( "       2    5    -7  -17  -31  -54  -18    0\r\n" );
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document.write( "That factors P(x) as\r\n" );
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document.write( "P(x) = (x - 1)(x - 1)(x - 1)(2x⁶ + 5x⁵ - 7x⁴- 17x³ - 31x² - 54x - 18)\r\n" );
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document.write( "Try -3 as a zero \r\n" );
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document.write( "    -3|2    5   -7  -17  -31  -54  -18 \r\n" );
document.write( "      |    -6    3   12   15   48   18 \r\n" );
document.write( "       2   -1   -4   -5  -16   -6    0\r\n" );
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document.write( "That factors P(x) as\r\n" );
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document.write( "P(x) = (x - 1)(x - 1)(x - 1)(x + 3)(2x⁵ - x⁴- 4x³ - 5x² - 16x - 6)\r\n" );
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document.write( "Try -3/2 as a zero \r\n" );
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document.write( "  -3/2|2   -1   -4  -5  -16  -6   \r\n" );
document.write( "      |    -3    6  -3   12   6\r\n" );
document.write( "       2   -4    2  -8   -4   0\r\n" );
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document.write( "That factors P(x) as\r\n" );
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document.write( "P(x) = (x - 1)(x - 1)(x - 1)(x + 3)(x + 3/2)(2x⁴- 4x³ + 2x² - 8x - 4)\r\n" );
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document.write( "We can factor 2 out of that last parentheses:\r\n" );
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document.write( "P(x) = (x - 1)(x - 1)(x - 1)(x + 3)(x + 3/2)(2)(x⁴- 2x³ + x² - 4x - 2)\r\n" );
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document.write( "P(x) = 2(x - 1)(x - 1)(x - 1)(x + 3)(x + 3/2)(x⁴- 2x³ + x² - 4x - 2)\r\n" );
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document.write( "Now we must factor the last polynomial:\r\n" );
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document.write( "x⁴- 2x³ + x² - 4x - 2\r\n" );
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document.write( "Rearrange as\r\n" );
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document.write( "x⁴+ x² - 2 - 2x³ - 4x\r\n" );
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document.write( "Factor -2x out of the last two terms:\r\n" );
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document.write( "(x⁴+ x² - 2) - 2x(x² + 2)\r\n" );
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document.write( "Factor the first parentheses:\r\n" );
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document.write( "(x² + 2)(x² - 1) - 2x(x² + 2)\r\n" );
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document.write( "Factor out (x² + 2)\r\n" );
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document.write( "(x² + 2)[(x² - 1) - 2x]\r\n" );
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document.write( "(x² + 2)(x² - 1 - 2x)\r\n" );
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document.write( "(x² + 2)(x² - 2x - 1)\r\n" );
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document.write( "Now we have factored P(x) as\r\n" );
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document.write( "P(x) = 2(x - 1)(x - 1)(x - 1)(x + 3)(x + 3/2)(x² + 2)(x² - 2x - 1)\r\n" );
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document.write( "P(x) = 2(x - 1)³(x + 3)(x + 3/2)(x² + 2)(x² - 2x - 1)\r\n" );
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document.write( "We set x² + 2 = 0\r\n" );
document.write( "           x² = -2 _\r\n" );
document.write( "            x = ±i√2\r\n" );
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document.write( "We set x² - 2x - 1 = 0\r\n" );
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document.write( "         _\r\n" );
document.write( "x = 1 ± √2\r\n" );
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document.write( "So, 1 is a zero of multiplicity 3,\r\n" );
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document.write( "3/2 is a zero\r\n" );
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document.write( "i√2  and -i√2  are zeros\r\n" );
document.write( "     _           _\r\n" );
document.write( "1 + √2  and 1 - √2  are zeros.\r\n" );
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document.write( "That's really a humdinger of a problem!\r\n" );
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document.write( "Edwin

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