document.write( "Question 1174391: Find a monic quartic polynomial f(x) with rational coefficients whose roots include x=1-sqrt(2) and x=2+sqrt(5). Give your answer in expanded form. I'm having trouble with this problem because I keep getting irrational coefficients. Can someone please help me? Thanks. \n" ); document.write( "

Algebra.Com's Answer #799829 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "Find a monic quartic polynomial f(x) with rational coefficients whose roots include x=1-sqrt(2) and x=2+sqrt(5).
\n" ); document.write( "Give your answer in expanded form.
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document.write( "Your monic quartic polynomial is p(x) = (x-(1-sqrt(2))*(x-(1+sqrt(2))*(x-(2+sqrt(5))*(x-(2-sqrt(5)).\r\n" );
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document.write( "It is simply the product of four binomials assosiated with the GIVEN roots and with theis conjugate roots \r\n" );
document.write( "over rational numbers.\r\n" );
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document.write( "The product of the first and the second binomials is\r\n" );
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document.write( "    (x-(1-sqrt(2))*(x-(1+sqrt(2)) = ((x-1)-sqrt(2))*((x-1)+sqrt(2)) = (x-1)^2 -4 = x^2 - 2x + 1 - 4 = x^2 - 2x -3.\r\n" );
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document.write( "The product of the third and the fourth binomials is\r\n" );
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document.write( "    (x-(1-sqrt(5))*(x-(1+sqrt(5)) = ((x-1)-sqrt(5))*((x-1)+sqrt(5)) = (x-1)^2 -25 = x^2 - 2x + 1 - 25 = x^2 - 2x -24.\r\n" );
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document.write( "The monic quartic polynomial is p(x) = (x^2 - 2x -3)*(x^2 - 2x -24) = x^4 - 4x^3 -23x^2 + 54x + 72.\r\n" );
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document.write( "It is your answer.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "When you complete reading my post, do not forget to post your \"THANK\" to me for my teaching.\r
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