I'm a little bit confused (technically I'm supposed to be using the program mathematica, but I need to know how to actually solve the problem first before I can type it in) The question says,
Find the cubic polynomial f(x) such that f(1)=f(2)=f(3)=1 and f(4)=7
I'm just really confused and not sure of what to do.. and I've never learned how to do this... For some reason I'm taking math 131 and the math lab 133 at the same time so I may be on here a lot... It's too late to drop too :/...
A polynomial is in the form \(\displaystyle f(x) = a_nx^{n} + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \). That said, a cubic polynomial is in the form \(\displaystyle f(x) = a_3x^{3} + a_2x^{2} + a_1x + a_0\).
Now, finding the solution to your problem is as simple as solving a system of equations. Consider this system:
\(\displaystyle
\begin{bmatrix}
1 & 1 & 1 & 1 & | & 1 \\
8 & 4 & 2 & 1 & | & 1 \\
27 & 9 & 3 & 1 & | & 1 \\
64 & 16 & 4 & 1 & | & 7 \\
\end{bmatrix} \begin{bmatrix} a_3 \\ a_2 \\ a_1 \\ a_0 \\ 1\end{bmatrix}
\)
It is simply an augmented matrix representing the system of equations that you described, i.e. the first row of Matrix 1 represents \(\displaystyle f(1): a_31^{3} + a_21^{2} + a_11 + a_0 = 1\), the second \(\displaystyle f(2)\) etc.
Mathematica should be able to solve this. If not in this form, in whatever form it accepts systems of equations. (E.g. a+b+c+d=1, 8a+4b+2c+d = 1, 27a+9b+3c+d=1, 64a+16b+4c+d=7)
(Edit: plugging this into http://www.wolframalpha.com does the trick.)
Otherwise, you could use
Gauss-Jordan Elimination to solve the above matrix.
Hope this helped. Just reply if you need any more help (i.e., you want me to walk through the Gauss-Jordan Elimination method to solve this matrix).
