document.write( "Question 816802: for what values of k will the remainder be the same when x^2 +kx +4 is divided by x-1 and x+1? \n" ); document.write( "

Algebra.Com's Answer #491731 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
One way to solve this is to divide by x-1 and x+1, set the remainders equal and solve for k. (I hope you've learned synthetic division because that is how I will divide. Long division will get the same result.)
\n" ); document.write( "

\r\n" );
document.write( "1  |   1   k    4\r\n" );
document.write( "----       1    k+1\r\n" );
document.write( "      -------------\r\n" );
document.write( "       1   k+1  k+5\r\n" );
document.write( "\r\n" );
document.write( "-1 |   1   k    4\r\n" );
document.write( "----       -1   -k+1\r\n" );
document.write( "      -------------\r\n" );
document.write( "       1   k-1  -k+5\r\n" );
document.write( "

k+5 = -k+5
\n" ); document.write( "Add k:
\n" ); document.write( "2k + 5 = 5
\n" ); document.write( "Subtract 5:
\n" ); document.write( "2k = 0
\n" ); document.write( "Divide by 2:
\n" ); document.write( "k = 0

\n" ); document.write( "Another way to solve this uses two key facts:

The Remainder Theorem tells us that the remainder is equal to the y-coordinate for the x-value that makes the divisor zero.
$\"x%5E2%2Bkx%2B4\"$ is a parabola when graphed. Parabolas are symmetric about the line through the vertex.

From the first fact and from the fact that the remainders are equal, we get that y-coordinate for x=1 (which makes (x-1) zero) and the y-coordinate for x = -1 (which makes (x+1) zero) are the same. From the symmetry of a parabola we get that x=1 and x=-1 can have the same y-coordinates only if they are equally distant from the vertex. Halfway between 1 and -1 is zero. So the x-coordinate has to be zero in order for x=1 and x=-1 to have the same y-coordinates.

\n" ); document.write( "And finally, in the general form for parabolas, $\"ax%5E2%2Bbx%2Bc\"$ , the x-coordinate of the vertex is given by $\"-b%2F2a\"$ . Since we have determined that this x-coordinate must be zero:
\n" ); document.write( " $\"0+=+-b%2F2a\"$
\n" ); document.write( "Since a fraction is equal to zero only if the numerator is zero, the only way this equation can be true is if b = 0. And since the \"b\" of $\"x%5E2%2Bkx%2B4\"$ is \"k\", k must be zero. \n" ); document.write( "

\n" );