document.write( "Question 825719: If x= sqrt(6) is a root of x^3-5x^2-6x+30=0, use synthetic division to factor the polynomial completely and list all real solutions of the equation. \n" ); document.write( "
Algebra.Com's Answer #497570 by jsmallt9(3758)\"\" \"About 
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document.write( "sqrt(6) |  1   -5           -6            30\r\n" );
document.write( "---------      sqrt(6)       6-5sqrt(6)  -30\r\n" );
document.write( "          ------------------------------------\r\n" );
document.write( "           1   -5+sqrt(6)   -5sqrt(6)      0\r\n" );
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The remainder is zero (in the lower right corner) which means our (x-sqrt(6)) divided evenly into our expression (as we knew it would since it is a root). The rest of the bottom row, \"1 -5+sqrt(6) -5sqrt(6)\", is the quotient.

\n" ); document.write( "If a polynomial with rational coefficients, like our expression, has a square root for a root, then the negative of that square root will also be a root. So we will now divide our quotient by (x-(-sqrt(6))):
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document.write( "-sqrt(6)|  1   -5+sqrt(6)   -5sqrt(6)\r\n" );
document.write( "---------        -sqrt(6)    5sqrt(6)\r\n" );
document.write( "          ------------------------------------\r\n" );
document.write( "           1   -5            0\r\n" );
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The quotient, \"1 -5\", translates into x-5. This makes the last root a 5.

\n" ); document.write( "So the roots of our expression are: 5, sqrt(6) and -sqrt(6) (which are all real numbers).

\n" ); document.write( "P.S. This is a polynomial, not a rational function. Please use an appropriate category when posting problems.
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