%% Polynomials

%% Working with Polynomials
% A polynomial in Matlab is represented by a row vector of coefficients.  The polynomial:
%
% $$x^4-7x^3+11x^2+7x-12$$
%
% is represented in Matlab as:

p = [ 1 -7 11 7 -12 ];

%% Find roots of the polynomial
% The Matlab command "roots" will provide the roots of a polynomials:

roots(p)

%% Define a polynomial by its roots
% We can build up a polynomial based on its roots:

r = [1 -1 4 3]
poly(r)

%% Multiply polynomials
% Polynomial multiplication corresponds to Matrix convolution, and so is done using the “conv” command in Matlab. For example, to compute:
%
% $$(x^2-3x+1)(x^5-x^3-1)$$
%
% Enter the Matlab commands:

p1 = [1 -3 1]
p2 = [1 0 -1 0 0 -1]
conv(p1,p2)

%% Polynomial Division
% Polynomial division corresponds to matrix deconvolution and is performed
% with the "deconv" command:
%
% $$(x^5-x^3+1)/(x^2-1)$$

num = [1 0 -1 0 0 1]
dem = [1 0 -1]
deconv(num, dem)

%% Deconv can return both quotient and remainder
[q, r] = deconv(num, dem)

%% Evaluate a polynomial at a point
% The Matlab function "polyval" will evaluate a polynomial at a point.

p
%%
polyval(p, 4)
%%
polyval(p, 5)
%%
polyval(p, [1 2 3 4 5])

%% Plot
x = -2:.1:5;
plot(x, polyval(p,x))

%% Exercises
% 1. Use 'polyder' to compute the derivative of a polynomial
%
% 2. Use 'polyint' to integrate a polynomial
%
% 3. Plot a polynomial and its derivative on the same axis
%
% 4. Find the critical points (i.e. roots of the derivative)
%
% 5. Format plots of the polynomial, its derivative, and/or the critical
% points