| The Third Degree |
The zeros of the polynomial
x3 - 33x2 + 354x + k
are in arithmetic progression.
What is the value of k? |
Problem Moderated by: Douglas |
| Problem Solution |
Let the three zeros be w, y, and z.
w = y - d and z = y + d, where d is the common difference of the progression
We know that the sum of the zeros is 33 (-b/a); thus:
w + y + z = 33
(y - d) + y + (y + d) = 33
3y = 33
y = 11
From this we also conclude that w + z = 22
Next, we know the sum of the zeros taken two at a time is 354 (c/a), so
11w + 11z + wz = 354
11(w + z) + wz = 354
11(22) + wz = 354
242 + wz = 354
wz = 112
Since k is the negative of the product of the zeros, k = -ywz = -11(112) = -1232
Note the 'shortcut' in this solution; you might make the assumption that you need to find all three zeros in order to find k. In fact, this is not the case, making the solution quicker and easier than expected.
Sasha further simplifies this by pointing out that once you've determined y = 11, you can simply use synthetic division to determine k; when you do the division, you obtain a remainder of k + 1232. Since the remainder is zero, k = -1232. |
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