<PRE><B>Question 1122472</B>
&lt;br&gt;
Given n points, a polynomial of degree (n-1) or less can fit the points.  With 5 points as in this problem, the polynomial that produces the 5 points will have degree 4 or less.&lt;br&gt;
We can use the method of finite differences to see if the polynomial might have degree less than 4.&lt;br&gt;&lt;pre&gt;
    48   0   -16   -18   0
      -48  -16   -2   18
         32   14    20
           -18    6
               24&lt;/pre&gt;&lt;br&gt;
We don&#39;t get a row of constant differences before the 4th row of differences, so the polynomial that produces the given points will be degree 4.&lt;br&gt;
The &quot;24&quot; in the 4th row of differences also tells us that the leading coefficient of the polynomial will be 24/(4!) = 24/24 = 1.&lt;br&gt;
The general polynomial of degree 4 is&lt;br&gt;
y = ax^4+bx^3+cx^2+dx+e&lt;br&gt;
Form a system of 5 equations in the 5 variables using the x and y value of the given points.  It will turn out that there are some shortcuts that make this particular system of equations relatively easy to solve.&lt;br&gt;&lt;pre&gt;
(1)   x=-2:    16a - 8b + 4c - 2d + e = 48
(2)   x = -1:    a -  b +  c -  d + e = 0
(3)   x = 0:                        e = -16
(4)   x = 1:     a +  b +  c +  d + e = -18
(5)   x = 2:   16a + 8b + 4c + 2d + e = 0&lt;br&gt;&lt;/pre&gt;
Adding equations (2) and (4) gives us&lt;br&gt;
{{{2a+2c+2e = -18}}}
{{{2a+2c-32 = -18}}}
{{{2a+2c = 14}}}
(6)  {{{a+c = 7}}}&lt;br&gt;
Adding equations (1) and (5) gives us&lt;br&gt;
{{{32a+8c+2e = 48}}}
{{{32a+8c-32 = 48}}}
{{{32a+8c = 80}}}
(7)  {{{4a+c = 10}}}&lt;br&gt;
Then solving equations (6) and (7) gives us a=1 (which we already knew) and c=6.&lt;br&gt;
Subtracting equation (2) from equation (4) gives us&lt;br&gt;
{{{2b+2d = -18}}}
(8)  {{{b+d = -9}}}&lt;br&gt;
Subtracting equation (1) from equation (5) gives us&lt;br&gt;
{{{16b+4d = -48}}}
(9)  {{{4b+d = -12}}}&lt;br&gt;
Then solving equations (8) and (9) gives us b = -1 and d = -8.&lt;br&gt;
So we have our coefficients:
a = 1; b = -1; c = 6; d = -8; e = -16&lt;br&gt;
The polynomial of least degree that passes through the 5 given points is&lt;br&gt;
{{{y = x^4-x^3+6x^2-8x-16}}}&lt;br&gt;
A graph that passes through (-2,48), (-1,0), (0,-16), (1,-18) and (2,0)...&lt;br&gt;
{{{graph(400,400,-3,3,-20,60,x^4-x^3+6x^2-8x-16)}}}
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