document.write( "Question 1014613: For the equation $\"x%5E4+%2B+x%5E3+-16x%5E2+-4x+%2B48=0\"$ the product of two of the roots is 6.\r
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Algebra.Com's Answer #630908 by Edwin McCravy(20064) $\"\"$ $\"About$

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document.write( "You are to express the 4th degree polynomial equation in this form:\r\n" );
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document.write( "Fact 1.  A polynomial equation has the same number of roots as \r\n" );
document.write( "its degree, including duplicate roots.\r\n" );
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document.write( "Fact 2: A monic polynomial is a polynomial with leading \r\n" );
document.write( "coefficient 1.\r\n" );
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document.write( "Fact 3:  The constant term of a monic polynomial is the product\r\n" );
document.write( "of the roots if the degree is even [and the negative of the\r\n" );
document.write( "product of the roots if the degree is odd].  \r\n" );
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document.write( "By fact 1,  has 4 roots.\r\n" );
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document.write( "by fact 3, the product of all four roots is 48.\r\n" );
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document.write( "We are told that the product of two of the roots is 6.\r\n" );
document.write( "Since we know that the product of ALL the roots is 48, and \r\n" );
document.write( "the product of two of them is 6, the product of the other \r\n" );
document.write( "two roots must be 48/6 = 8\r\n" );
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document.write( "Let  be the quadratic whose two roots have \r\n" );
document.write( "product 6.  Then by the fact 3 above, b=6\r\n" );
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document.write( "Let  be the quadratic whose two roots have \r\n" );
document.write( "product 8.  Then by the fact 3 above, d=8.\r\n" );
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document.write( "So we have the identity:\r\n" );
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document.write( "Multiply out the right side:\r\n" );
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document.write( "Equate the terms in \r\n" );
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document.write( "Divide through by \r\n" );
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document.write( "Equate the terms in \r\n" );
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document.write( "Divide through by \r\n" );
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document.write( "Equate the terms in \r\n" );
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document.write( "Divide through by \r\n" );
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document.write( "So we have the three equations:\r\n" );
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document.write( "Solve any two of those by substitution or elimination\r\n" );
document.write( "and get a=-5 and c=6.\r\n" );
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document.write( "So  in the form\r\n" );
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document.write( "is  \r\n" );
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document.write( "To find the roots of the equation, factor each of those\r\n" );
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document.write( "Set each factor = 0 and the roots are:\r\n" );
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document.write( "2, 3, -4, and -2\r\n" );
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document.write( "Edwin

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