With k = 2 the equation becomes:
If point B lies on the curve then its coordinates must fit the equation. So:
Now we just solve this for p.
The first thing we must do is get one side to be zero. Subtracting 35 we get:
Next we try to factor the right side. The greatest common factor (GCF) is 1 (which we rarely bother factoring out). There are too many terms for factoring by patterns or for trinomial factoring. And I do not see how to factor by grouping. All that's left is trial and error of the possible rational roots.
The possible rational roots of a polynomial are all the possible ratios, positive and negative, made with a factor of the constant term (at the end) over a factor of the leading coefficient (at the front). Our constant term is 3 whose factors are 1 and 3. (Actually it is -3 but since we will try all positive and negative ratios we can just as well use 3.) And the leading coefficient (in front ) is 1 whose factors are just 1's. This makes the possible rational roots:
+1/1 and +3/1
which simplify to:
+1 and +3
So there are 4 possible rational roots.
Since you've already figured out that it should be near -4, we'll try -3 first. Checking to see if a possible root is a root is more easily done with synthetic division:
-3 | 1 -2 -16 -3
---- -3 15 3
------------------
1 -5 -1 0
The remainder, in the lower right corner, is zero. This means that (x - (-3)) is a factor and that -3 is a root of the polynomial. So p = -3 or the coordinates of B are (-3. 35).