ooooh, I like these. Divide x^4+kx^3+2 by x+1... and the other factor has be a whole polynomial. Start by adding in the other orders of x: x^4 + kx^3 + 0*x^2 + 0*x +2. Now, using long division, divide that by x + 1. (in order to understand my next comments, begin writing the long division problem and start to solve it). The first term on top is now x^3. Now, Be carefuly to treat k properly. kx^3 - x^3 = (k-1)x^3. Drop down the 0*x^2, and multiply by something that gets rid of the x^3 term... namely, (k-1)x^2. The x^3 term drops by subtraction, and 0*x^2 - (k-1)x^2 = (1-k)x^2... and drop down the 0*x term. Now, get rid of the x^2 term by multiplying by (1-k)x. Subtract 0*x - (1-k)x = (k-1)x... and bring down the last value: 2. Here's where we find out what k is. We're left with the division problem: (k-1)x + 2 divided by (x+1) has to be a whole division. That means that the x term and the 2 term will vanish after the subtration. That means the next number up top is 2. Multiply, and subtract: (k-1)x - 2x will be zero. 2 - 2 will also be zero. BUt, we can now solve for k with (k-1) - 2 = 0. k - 1 - 2 = 0 k - 3 = 0 k = 3 It's hard to 'describe' long division of polynomials in text, but I hope this helps.

x = -1 is a root So let's calculate for this value: (-1)^4 +k(-1)^3 + 2 = 0 1 - k + 2 = 0 k = 3