document.write( "Question 77338: Polynomial function:
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Algebra.Com's Answer #55463 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "Polynomial function:\r\n" );
document.write( "a. List all possible zeros\r\n" );
document.write( "b. find all rational zeros \r\n" );
document.write( "c. list all factors! \r\n" );
document.write( "f(x) = x³ + 5x² + 2x - 8\r\n" );
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document.write( "The rule is: If a polynomial arranged in descending order\r\n" );
document.write( "has any rational zeros, then they will be among the set \r\n" );
document.write( "of positive and negative fractions, each of whose numerator \r\n" );
document.write( "is a factor of the constant term, (i.e. the term with no\r\n" );
document.write( "vraiable), and whose denominator is a factor of the \r\n" );
document.write( "coefficient of the first term.\r\n" );
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document.write( "The polynomial is so arranged.\r\n" );
document.write( "The constant term is -8, and the coefficient of the first\r\n" );
document.write( "term is 1, so if there are any rational zeros, they will\r\n" );
document.write( "be among these:\r\n" );
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document.write( "±1, ±2, ±4, ±8\r\n" );
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document.write( "Try the easiest one first, 1. If 1 is a solution, then\r\n" );
document.write( "(x - 1) will be a factor of f(x)\r\n" );
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document.write( "using synthetic division we divide by (x - 1):\r\n" );
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document.write( "1|1  5  2 -8\r\n" );
document.write( " |   1  6  8\r\n" );
document.write( "  1  6  8  0\r\n" );
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document.write( "Since the last number on the bottom row of the\r\n" );
document.write( "aynthetic division (the remainder) is 0, we\r\n" );
document.write( "have factored f(x) as\r\n" );
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document.write( "f(x) = (x - 1)(x² + 6x + 8)\r\n" );
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document.write( "We can now factor the trinomial in the second\r\n" );
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document.write( "f(x) = (x - 1)(x + 2)(x + 4)\r\n" );
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document.write( "That's the list of factors, (x - 1), (x + 2), and\r\n" );
document.write( "(x + 4)\r\n" );
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document.write( "Setting each of the factors = 0 we have\r\n" );
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document.write( "x = 1, x = -2, and x = -4 as the three zeros.\r\n" );
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document.write( "Here's the graph of f(x). Notice that it crosses\r\n" );
document.write( "the x-axis 3 times, once at each those three zeros. \r\n" );
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document.write( "Edwin

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