document.write( "Question 1081177: Let f(x)=3x^4 + 7x^3 + ax^2 + bx -14 where a and b are constants.If (x-1) is a factor of f(x) and when f(x) is divided by (x+1), the remainder is -12, find the values of a and b. With these values of a and b,\r
\n" ); document.write( "\n" ); document.write( "(A) find a factor of f(x) in the form x+k where k is a postive integer.
\n" ); document.write( "(B) write f(x) in the form
\n" ); document.write( " f(x)=(x-1)(x+k)Q(x),where Q(x)is a real quadratic.\r
\n" ); document.write( "\n" ); document.write( "Hence,show that Q(x) is irreducible. \n" ); document.write( "

Algebra.Com's Answer #695237 by josgarithmetic(39625) $\"\"$ $\"About$

You can put this solution on YOUR website!
x+1 is a factor.\r
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( " 1    |    3   7   a   b    -14\r\n" );
document.write( "      |\r\n" );
document.write( "      |---------------------------\r\n" );
document.write( "\r\n" );
document.write( "         3  10   10+a  b+a+10  a+b-4\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Remainder must be equal to zero.\r\n" );
document.write( "\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Division by x+1 gives remainder of -12.\r
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( " -1   |    3   7   a   b    -14\r\n" );
document.write( "      |\r\n" );
document.write( "      |---------------------------\r\n" );
document.write( "          3  4   a-4  b-a+4  a-b-18\r\n" );
document.write( "\r\n" );
document.write( "Remainder is given as -12.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "System to solve for a and b: $\"system%28a%2Bb=4%2Ca-b=6%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"system%28a=5%2Cb=-1%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the cubic result for the first synthetic division, you have $\"%28x-1%29%283x%5E3%2B10x%5E2%2B15x%2B14%29\"$ . If you do synthetic division checking $\"x%2B2\"$ or root of $\"-2\"$ , you will find remainder 0, meaning $\"x%2B2\"$ is also a root.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=%28x-1%29%28x%2B2%29%283x%5E2%2B4x%2B7%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---- \n" ); document.write( "

\n" );