# how to solve for the equation if x = 2001, 2002, 2003, 2004, 2005 and the f(x) = 2677, 2685, 2696, 3120, 3154 respectively?

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The problem is ambiguous since there is no indication about the order of the equation that represents the function.

Since the problem gives 5 values for x and 5 values for f(x), you may assume that the equation of the function is a fourth order polynomial such that:

`f(x) = ax^4 + bx^3 + cx^2 + dx + e`

`f(2001) = 2677 =gt a(2001)^4 + b(2001)^3 + c(2001)^2 + 2001d + e = 2677`

`f(2002) = 2685 =gt a(2002)^4 + b(2002)^3 + c(2002)^2 + 2002d + e = 2685`

`f(2003) = 2696 =gt a(2003)^4 + b(2003)^3 + c(2003)^2 + 2003d + e = 2696`

`f(2004) =3120 =gt a(2004)^4 + b(2004)^3 + c(2004)^2 + 2004d + e = 3120`

`f(2005) = 3154=gt a(2005)^4 + b(2005)^3 + c(2005)^2 + 2005d + e = 3154`

You need to evaluate the determinant of the system of equations such that:

`Delta = [[(2001)^4,(2001)^3,(2001)^2,(2001)^1,1],[(2002)^4,(2002)^3,(2002)^2,(2002)^1,1],[(2003)^4,(2003)^3,(2003)^2,(2003)^1,1],[(2004)^4,(2004)^3,(2004)^2,(2004)^1,1],[(2005)^4,(2005)^3,(2005)^2,(2005)^1,1]]`

This determinant is called Vandermonde's determinant and it is evaluated such that:

`Delta = (2002-2001)(2003-2001)(2004-2001)(2005-2001)(2003-2002)(2004-2002)(2005-2002)(2004-2003)(2005-2003)(2005-2004)`

`Delta = 1*2*3*4*1*2*3*1*2*1`

`Delta = 2^3*3^2*4 =gt Delta = 288`

Since `Delta != 0 =gt ` the system may be solved using Cramer's solutions such that:

`a = (Delta_a)/Delta ; b = (Delta_b)/Delta ; c = (Delta_c)/Delta ; d = (Delta_d)/Delta ; e = (Delta_e)/Delta`

`Delta_a = [[2677,(2001)^3,(2001)^2,(2001)^1,1],[2685,(2002)^3,(2002)^2,(2002)^1,1],[2696,(2003)^3,(2003)^2,(2003)^1,1],[3120,(2004)^3,(2004)^2,(2004)^1,1],[3154,(2005)^3,(2005)^2,(2005)^1,1]]`

`Delta_b = [[(2001)^4,2677,(2001)^2,(2001)^1,1],[(2002)^4,2685,(2002)^2,(2002)^1,1],[(2003)^4,2696,(2003)^2,(2003)^1,1],[(2004)^4,3120,(2004)^2,(2004)^1,1],[(2005)^4,3154,(2005)^2,(2005)^1,1]]`

Notice that you may evaluate `Delta_c,` substituting the column of coefficients of variable c by the column of constant terms, like the previous examples `Delta_a` and `Delta_b` prove. The same you can do for the rest of determinants, `Delta_d` and `Delta_e` .

**Hence, supposing that the equation of function you need to find is a polynomial of fourth order, you may evaluate the coefficients a,b,c,d and e, using Cramer's rule.**