Finding the value of k in ...?

Question

In the polynomial (x^3 + kx + 4x +1) / (x +1), how do I use synthetic division to find the value of k so that the remainder is zero?

** Sorry, I meant to write kx^2 above

Answer 1

Don't you mean kx²? I'll assume you do, given the form, but if you actually meant to write x³+(k+4)x+1, this isn't going to be the right answer.

First, find the remainder:

-1 | 1 k .....4 ......1
........ -1...1-k ... k-5
--------------------------
......1 k-1 5-k.....k-4

So the remainder is k-4. If the remainder is 0, then k-4=0 and k=4.

Answer 2

:Hopefully you know how to do synthetic division as this is what it would look like: (assuming the second term is actually squared)

-1 | 1 k 4 1
| -1 -k +1 k - 5
|____________________
1 k - 1 5 - k k - 4

The last expression, k - 4, is the remainder. Set this equal to zero and you can see that k = 4 for there to be no remainder.

I wouldn't do the question this way though....remainder theorem would be easier.

P(-1) = 0
-1 + k - 4 + 1 = 0
k = 4