Question 714127
With k = 2 the equation becomes:
{{{y = x^3 - 2x^2 - 16x + 32}}}
If point B lies on the curve then its coordinates must fit the equation. So:
{{{35 = p^3 - 2p^2 - 16p + 32}}}
Now we just solve this for p.<br>
The first thing we must do is get one side to be zero. Subtracting 35 we get:
{{{0 = p^3 - 2p^2 - 16p - 3}}}
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.<br>
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 {{{p^3}}}) is 1 whose factors are just 1's. This makes the possible rational roots:
<u>+</u>1/1 and <u>+</u>3/1
which simplify to:
<u>+</u>1 and <u>+</u>3
So there are 4 possible rational roots.<br>
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:
<pre>
-3 |   1   -2   -16   -3
----       -3    15    3
      ------------------
       1   -5    -1    0
</pre>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).<br>