Create vectors u and v containing the coefficients of the polynomials $$x^2+1$$ and $$2x+7$$.

u = [1 0 1];
v = [2 7];

Use convolution to multiply the polynomials.

w = conv(u,v)

w = 1×4

     2     7     2     7

w contains the polynomial coefficients for $$2x^3+7x^2+2x+7$$.

Create two vectors and convolve them.

u = [1 1 1];
v = [1 1 0 0 0 1 1];
w = conv(u,v)

w = 1×9

     1     2     2     1     0     1     2     2     1

The length of w is length(u)+length(v)-1, which in this example is 9.

Create two vectors. Find the central part of the convolution of u and v that is the same size as u.

u = [-1 2 3 -2 0 1 2];
v = [2 4 -1 1];
w = conv(u,v,'same')

w = 1×7

    15     5    -9     7     6     7    -1

w has a length of 7. The full convolution would be of length length(u)+length(v)-1, which in this example would be 10.