document.write( "Question 1183420: Pls help me get remainder when 2x^4 - 5x^3 + 7x^2 - x + 6 is divided by x^2+x-2 by using only remainder theorem. Thnx. \n" ); document.write( "

Algebra.Com's Answer #813736 by ikleyn(52814) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Pls help me get remainder when 2x^4 - 5x^3 + 7x^2 - x + 6 is divided by x^2+x-2 by using only remainder theorem.
\n" ); document.write( "~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            @MathLover1 solved the problem, but used the long division of polynomials\r
\n" ); document.write( "\n" ); document.write( "            instead of the Remainder theorem, as it was requested in the post.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            Below I developed the solution to this problem using the Remainder theorem - - - as requested.         E N J O Y ( ! )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Since the divisor  (x^2 + x - 2)  is the quadratic polynomial, the reminder of the division is the linear binomial\r\n" );
document.write( "and, therefore, has the form  (ax+b).\r\n" );
document.write( "\r\n" );
document.write( "Our task is to find / (to determine) the coefficients \"a\" and \"b\" of this linear binomial.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we have\r\n" );
document.write( "\r\n" );
document.write( "    2x^4 - 5x^3 + 7x^2 - x + 6 = q(x)*(x^2+x-2) + (ax+b).     (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "where q(x) is another quadratic polynomial, but I even will not touch it . . . \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "Now notice that the roots of the quadratic polynomial (x^2+x-2) are -2 and 1\r\n" );
document.write( "(because  x^2+x-2 = (x+2)*(x-1) in the factored form).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The remainder of division the quartic polynomial  2x^4 - 5x^3 + 7x^2 - x + 6  by  (x+2)  is the value of this quartic \r\n" );
document.write( "at x= -2 (the Remainder theorem), i.e.\r\n" );
document.write( "\r\n" );
document.write( "    2*((-2)^4) - 5*((-2)^3) + 7*((-2)^2) - (-2) + 6 =  108.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Due to equality (1), it means that the linear binomial (ax+b)   has the value of  108  at  x= -2,  i.e.\r\n" );
document.write( "\r\n" );
document.write( "    -2a + b = 108     (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The remainder of division the quartic polynomial  2x^4 - 5x^3 + 7x^2 - x + 6  by  (x-1)  is the value of this quartic \r\n" );
document.write( "at x= 1 (the Remainder theorem), i.e.\r\n" );
document.write( "\r\n" );
document.write( "    2*(1^4) - 5*(1^3) + 7*(1^2) - 1 + 6 =  9.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Due to equality (1), it means that the linear binomial (ax+b)   has the value of 9 at x= 1, i.e.\r\n" );
document.write( "\r\n" );
document.write( "    a + b = 9         (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "   +-------------------------------------------------------------------------+\r\n" );
document.write( "   | Thus we have two equations (2) and (3) to find two unknowns \"a\" and \"b. |\r\n" );
document.write( "   +-------------------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To solve the system, subtract equation (3) from equation (2) to get\r\n" );
document.write( "\r\n" );
document.write( "    -2a - a = 108-9,   -3a = 99,  a = 99/(-3) = -33.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then from (3) you get  b= 9 - a = 9 - (-33) = 9 + 33 = 42.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus the problem is just solved (by the method you requested), and the ANSWER is:\r\n" );
document.write( "\r\n" );
document.write( "    +---------------------------------------------------------------+\r\n" );
document.write( "    |    the remainder of division of  2x^4 - 5x^3 + 7x^2 - x + 6   |\r\n" );
document.write( "    |              by  (x^2+x-2)  is  -33x + 42.                    |\r\n" );
document.write( "    +---------------------------------------------------------------+\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved as requested and carefully explained.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );