document.write( "Question 1157927: Determine $\"%28r+%2B+s%29%28s+%2B+t%29%28t+%2B+r%29\"$ , if r, s, and t are the three real roots of the polynomial $\"x%5E3+%2B+9x%5E2+-+9x+-+8\"$ . \n" ); document.write( "

Algebra.Com's Answer #780826 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "Determine $\"%28r%2Bs%29%28s%2Bt%29%28r%2Bt%29\"$ , if r, s, and t are the three real roots of the polynomial $\"x%5E3%2B9x%5E2-9x-8\"$ .
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document.write( "(r+s)(s+t)(r+t) = ((r + s + t)-t) * ((s + t + r) - r) * ((r + t + s) - s) =\r\n" );
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document.write( "    In the last three factors, replace  r+s+t by -9 (the value opposite to the coefficient at x^2).\r\n" );
document.write( "    Then continue the equality\r\n" );
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document.write( "= (-9-t)*(-9-r)*(-9-s) = -(9+t)*(9+r)*(9+s) = \r\n" );
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document.write( "= -(81 + 9t + 9r + tr)*(9+s) = -(729 + 81t + 81r + 9tr + 81s + 9ts + 9rs + trs) = \r\n" );
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document.write( "= -(729 + 81*(t + r + s) + 9*(tr + ts + rs) + trs) = \r\n" );
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document.write( "    In the last expression, replace  (t+r+s) by -9 (the value opposite to the coefficient at x^2);\r\n" );
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document.write( "    replace  (tr + ts + rs) by -9 (the value of the coefficient at x),\r\n" );
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document.write( "    and replace trs by 8 (the value opposite to the coefficient at the constant term of the polynomial, by Vieta's theorem).  \r\n" );
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document.write( "    You can continue then in this way\r\n" );
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document.write( "= -(729 + 81*(-9) + 9*(-9) + 8) = 73.     ANSWER\r\n" );
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