Transforming from Maclaurin to Taylor

I have a polynomial function in terms of x, in which the highest power never exceeds 6. This function will look like a Maclaurin Series with finite terms. How to tell Matlab to rewrite this function into a Taylor one, which will express the function in terms of (x-a), where a is a real, positive number?

E.g. Original function: f(x)=x^3+x^2+x+1

Transformed functions:

If a=1, g(x)=(x-1)^3+4*(x-1)^2+6*(x-1)+4

if a=2, g(x)=(x-2)^3+7*(x-2)^2+17*(x-2)+15

and etc.

I tried using rem but it returned an error message, saying that f(x) must be a real value in order to use rem.

Answer 1

[William Alberg]

If i understand correctly, your code looks something like this:

syms x

f(x) = x^3 + x^2 + x +1
f(x-1)
f(x-2)

And you want to expand the expressions. This can be done with the expand command:

expand(f(x-1))
>> x^3 - 2*x^2 + 2*x
expand(f(x-2))
>> x^3 - 5*x^2 + 9*x - 5

[Angus Wong]

Sorry, I didn't mean that. I mean like I have a polynomial in terms of x. I want Matlab to rewrite my expression in terms of (x-a) instead of x, where a is a positive number. The rewritten function will be the same as the original function when plotted on a graph. Transforming a Maculan series into a Tayler one is what I am asking for help.

[William Alberg]

Ahh, I completely misunderstood that!

I dont know how to achieve that, since matlab dont like the "k*(x-a)^1" part, and will rewrite it. I can however get the correct "k*(x-a)^n" for n > 1

My (not complete) solution is this:

f(x) = x^3 + x^2 + x +1;

a = 1;
temp = expand(subs(f(x),x,y + a))
g(x) = subs(temp,y,x-a)
>>6*x + 4*(x - 1)^2 + (x - 1)^3 - 2

[Angus Wong]

I appologise again for my vague wordings - I have no idea what I was typing earlier. Your new solution do help, thanks a lot!