Find the cubic polynomial satisfying f(0) = -5, f(1) = 0, f(2) = 15, f(3) = 52.?

[escobar147]

Here is the correct answer: 2x^3 - x^2 + 4x - 5

My attempt only gives me one cubed term and the other terms are also marginally off, any help on who can show me how to get the correct answer will be hugely appreciated

 

[Vargo]

Howdy,

What you need are Lagrange interpolation polynomials:
http://en.wikipedia.org/wiki/Lagrange_polynomial

 

[Norwegian]

Hi there,

What you really need is to translate the problem into a set of linear equations. If you let f(x)=ax³+bx²+cx+d, then f(0)=-5 gives d=-5, f(1)=0 gives a+b+c+d=0 etc, and you can quickly find the correct values of a, b, c and d.

 

[Vargo]

I guess it is a matter of taste as to whether one prefers to solve a system of linear equations or to just write down the answer.

 

[CellCoree]

It is important to note that there is not just one correct answer for this problem. There are multiple cubic polynomials that could satisfy the given conditions. However, the polynomial you provided, 2x^3 - x^2 + 4x - 5, does indeed satisfy the given conditions.

To find this polynomial, we can start by creating a system of equations using the given values for f(0), f(1), f(2), and f(3). We can then solve for the coefficients of the polynomial.

f(0) = -5 = a(0)^3 + b(0)^2 + c(0) + d
f(1) = 0 = a(1)^3 + b(1)^2 + c(1) + d
f(2) = 15 = a(2)^3 + b(2)^2 + c(2) + d
f(3) = 52 = a(3)^3 + b(3)^2 + c(3) + d

Simplifying these equations, we get:
-5 = d
0 = a + b + c + d
15 = 8a + 4b + 2c + d
52 = 27a + 9b + 3c + d

Solving this system of equations, we get a = 2, b = -1, c = 4, and d = -5. Thus, the cubic polynomial that satisfies the given conditions is 2x^3 - x^2 + 4x - 5.

If you are getting a different polynomial as your solution, it is possible that there was a mistake in your calculations or in setting up the system of equations. It may be helpful to double check your work or to seek assistance from a tutor or colleague.