Question

if i have a known points of polynomial(2,5),(3,2) and (4,5). what would be the polynomial function will be?

Answer 1

I think you mean "if the points (2,5), (3,2) and (4,5) satisfy an unknown 3rd degree polynomial, what is the polynomial?"

Since 3 roots {2, 3, 4} are known, we might begin by assuming that this poly would have the form y = ax^3 + bx^2 +  cx + d (which has three factors).  Unfortunately, three roots are not enough to determine all four constants {a, b, c, d}.

So, let's assume, instead, that the poly would have the form y = ax^2 + bx + c.  Three given points should make it possible to determine {a, b, c}.

(2,5):  5 = a(2)^2 + b(2) + c => 5 = 4a + 2b + c

(3,2):  2 = a(3)^2 + b(3) + c   => 2 = 9a + 3b + 5 - 4a - 2b

(4,5):  5 = a(4)^2 + b(4) + c   => 5 = 16a + 4b + 5 - 4a - 2b

Now we have two equations in a and b alone, which enables us to solve for a and b:

2 = 9a + 3b + 5 - 4a - 2b becomes -3 = 5a + b
and 
5 = 16a + 4b + 5 - 4a - 2b becomes 0 = 12a + 2b, or 0 = 6a + b, or 0=-6a-b

Adding this result to -3 = 5a + b, we get -3 = -a, so a =3.
Thus, since -3 = 5a + b, -3 = 5(3) + b, so b = -18

All we have to do now is to find c.  Let's do this using 5 = 4a + 2b + c.
We know that a = 3 and b = -18, so  this becomes   5 = 4(3) + 2(-18) + c.

Thus, 5 = 12 - 36 + c, or c = 29.

With a, b and c now known, we can write the poly as y = 3x^2 - 18x + 29.

Now the only thing to do remaining is to verify that each of the three given points satsifies        y = 3x^2 - 18x + 29.  Try this, please.