document.write( "Question 43914This question is from textbook Algebra and Trigonometry with Analytic Geometey
\n" ); document.write( ": The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3+i among its roots. Express f(x) as a product of linear and quadratic polynomials polynomials with real coefficients. \r
\n" ); document.write( "\n" ); document.write( "This question I'm having trouble with but I came up witha an answer of; f(x)=(x-4)(x^2-6x-9) is this right. thanks for looking at this question. \n" ); document.write( "
Algebra.Com's Answer #28812 by AnlytcPhil(1807)\"\" \"About 
You can put this solution on YOUR website!
The degree three polynomial f(x) with real coefficients and\r\n" );
document.write( "leading coefficient 1, has 4 and 3+i among its roots. \r\n" );
document.write( "Express f(x) as a product of linear and quadratic polynomials\r\n" );
document.write( "polynomials with real coefficients. This question I'm having \r\n" );
document.write( "trouble with but I came up witha an answer of:\r\n" );
document.write( "f(x)=(x-4)(x^2-6x-9) is this right. thanks for looking at \r\n" );
document.write( "this question.\r\n" );
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document.write( "No that's not correct.  You must know the following facts about\r\n" );
document.write( "polynomials:\r\n" );
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document.write( "1. A degree n polynomial has n roots, counting multiplicities.\r\n" );
document.write( "2. If r is a root of a polynomial then (x-r) is a factor of the\r\n" );
document.write( "   polynomial.\r\n" );
document.write( "3. If a polynomial with real coefficients has p+qi as one root,\r\n" );
document.write( "   it also has its conjugate p-qi as another root.\r\n" );
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document.write( "A, By 2 above, since 4 is a root then (x-4) is a factor.\r\n" );
document.write( "         (You did this)\r\n" );
document.write( "B. By 2 above, since 3+i is a root then [x-(3+i)] is a factor\r\n" );
document.write( "   of the polynomial. \r\n" );
document.write( "C. By 3 above, since 3+i is a root then 3-1 is a factor of the\r\n" );
document.write( "   polynomial.\r\n" );
document.write( "D. By C and 2, since 3-i is a root then [x-(3-i)] is a factor \r\n" );
document.write( "   of the polynomial.\r\n" );
document.write( "E. By 1, there are no more roots besides 4, 3+i, and 3-1\r\n" );
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document.write( "Therefore f(x) is the product of the three factors. It will \r\n" );
document.write( "have leading coefficient 1 because the c oefficients of x in \r\n" );
document.write( "all three factors is 1.\r\n" );
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document.write( "[Note: If it had had a leading coefficient other than 1, we \r\n" );
document.write( " would have to multiply the polynomial by it, too, but thet \r\n" );
document.write( " is unnecessary here.]\r\n" );
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document.write( "So we have:\r\n" );
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document.write( "f(x) = (x - 4)[x - (3+i)][x - (3-i)]\r\n" );
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document.write( "But the two factors containing imaginary numbers have to be\r\n" );
document.write( "multiplied together, since all coefficients must be real.\r\n" );
document.write( "We can use FOIL to multiply the last two factors together:\r\n" );
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document.write( "f(x) = (x - 4)[x - (3+i)][x - (3-i)]\r\n" );
document.write( "f(x) = (x - 4)[x² - (3-i)x - (3+i)x + (3+i)(3-i)]\r\n" );
document.write( "f(x) = (x - 4)[x² - (3x-ix) - (3x+ix) + (9-3i+3i-i²)]\r\n" );
document.write( "f(x) = (x - 4)[x² - 3x + ix - 3x - ix + (9-i²)]\r\n" );
document.write( "f(x) = (x - 4)[x² - 6x + 9 - i²]\r\n" );
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document.write( "Now since i² = -1 we substitute (-1) for i².\r\n" );
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document.write( "f(x) = (x - 4)[x² - 6x + 9 - (-1)]\r\n" );
document.write( "f(x) = (x - 4)[x² - 6x + 9 + 1]\r\n" );
document.write( "f(x) = (x - 4)[x² - 6x + 10]\r\n" );
document.write( "f(x) = (x - 4)(x² - 6x + 10)\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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