I am trying to work with a $5^{th}$ degree polynomial, and I am using Excel to calculate a load of mathematical formulas and equations. I have the form
$$f(t)=a\,(1-t)^5+b\,t(1-t)^4+c\,t^2(1-t)^3+d\,t^3(1-t)^2+e\,t^4(1-t)+f\,t^5$$
The problem is that I am using a complex solver in Excel for solving polynomials, and it can not be in this form. I need to have the equation simplified. I know how to do it by hand, but I am working with hundreds of equations and would rather keep my sanity. It was easy to work with $3^{rd}$ degrees because I had the following:
$$x(t) = a_x t^3 + b_x t^2 + c_x t + x_0 \\ \begin{align} & \quad x_1 = x_0 + c_x / 3 \\ & \quad x_2 = x_1 + (c_x + b_x) / 3 \\ & \quad x_3 = x_0 + c_x + b_x + a_x \\ \end{align} $$
$$ y(t) = a_y t^3 + b_y t^2 + c_yt + y_0 \\ \begin{align} & \quad y_1 = y_0 + c_y / 3 \\ & \quad y_2 = y_1 + (c_y + b_y) / 3 \\ & \quad y_3 = y_0 + c_y + b_y + a_y \\ \end{align} $$
From there I could complete my own calculations and place in Excel for what I needed. But I am unsure how to do this for a $5^{th}$ degree. Is there a simple formula like the one above for finding $x_0,x_1,x_2,x_3,x_4,x_5$ and $y_0,y_1,y_2,y_3,y_4,y_5$?
I appreciate any help, it has been 10 years since I took a math class and my mind is already overloaded by learning these higher degrees. So any terminology or hints given please keep that in mind. (I have already searched for hours on Google to no avail) Thanks in advance!
I figured out a way that works for what I need:
$$\ f(t) = (-a+5b-10c+10d-5e+f)t^5+(5a-20b+30c-20d+5e)t^4+(-10a+30b-30c+10d)t^3+(10a-20b+10c)t^2+(-5a+5b)t+a $$
Where a-f are the control points to the curve. This allows me to work with the formulas in excel the way I need to.