document.write( "Question 1172086: Find the equation of a degree 4 polynomial with zeros x=5, x=−6, and x=1+i. It passes through the point (1,84). Write the equation in general form.\r
\n" );
document.write( "\n" );
document.write( "I'm just not sure about how to use the complex roots to find the equation, since there would have to be a conjugate and the two conjugates multiplied together just make 2 (1+i)(1-i)=-i^2+1=+1+1=2\r
\n" );
document.write( "\n" );
document.write( "Thanks for your help! \n" );
document.write( "
Algebra.Com's Answer #796999 by ikleyn(52805)![]() ![]() You can put this solution on YOUR website! .\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " Watch attentively every my step.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \r\n" ); document.write( "The polynomial is ASSUMED to be with real coefficients - it is the condition you missed in your post (!)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Then the roots are 5, -6, 1+i and 1-i (together with 1+i, its conjugate 1-i is the root, too - for such a polynomial).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Then the associate linear binomials are \r\n" ); document.write( "\r\n" ); document.write( " (x-5), (x+6), (x-(1+i)) and (x-)1-i)).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "We seek for the product of these binomials with the unknown real coefficient \"a\"\r\n" ); document.write( "\r\n" ); document.write( " f(x) = a*(x-5)*(x+6)*(x-(1+i))*(x-(1-i)).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "The product of the last two binomials is\r\n" ); document.write( "\r\n" ); document.write( " (x-(1+i))*(x-(1-i)) = x^2 - 2x + (1+i)*(1-i) = x^2 - 2x +2.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Therefore, the polynomial f(x) is\r\n" ); document.write( "\r\n" ); document.write( " f(x) = a*(x-5)*(x+6)*(x^2 - 2x + 2)\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "To find \"a\", use the fact that f(1) = 84. It gives\r\n" ); document.write( "\r\n" ); document.write( " 84 = a*(1-5)*(1+6)*(1 - 2 + 2) = a*(-4)*7*1 = -28a.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Hence, a= 84/(-28) = -4.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "So, your polynomial is\r\n" ); document.write( "\r\n" ); document.write( " f(x) = -4*(x-5)*(x+6)*(x^2 - 2x + 2).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "It is your ANSWER. \r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "You may transform it further, if you want / (if you need).\r\n" ); document.write( "\r \n" ); document.write( "\n" ); document.write( "Solved and explained.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |