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.)
1 | 1 k 4
---- 1 k+1
-------------
1 k+1 k+5
-1 | 1 k 4
---- -1 -k+1
-------------
1 k-1 -k+5
k+5 = -k+5
Add k:
2k + 5 = 5
Subtract 5:
2k = 0
Divide by 2:
k = 0
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.
- 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.
And finally, in the general form for parabolas, , the x-coordinate of the vertex is given by . Since we have determined that this x-coordinate must be zero:
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 is "k", k must be zero.