A polynomial #P(x)# has a zero #x_0# if and only if #(x-x_0)# is a factor of #P(x)#. Using that, we can work backwards to make a polynomial with given zeros by multiplying each necessary factor of #(x-x_0)#.
As our desired polynomial has #0# and #10# as zeros, it must have #(x-0)# and #(x-10)# as factors. Multiplying these, we get
#(x-0)(x-10) = x(x-10) = x^2-10x#
This is a polynomial of least degree which has #0# and #10# as zeros. Note that multiplying this by any other polynomial or constant will also result in a polynomial with #0# and #10# as zeros.
