After several operations of differentiation and multiplying by (x + 1) performed in an arbitrary order the polynomial x^8 + x^7 is changed to ax + b. Prove that the difference between the integers a and b is always divisible by 49.
After several operations of differentiation and multiplying by (x + 1) performed in an arbitrary order the polynomial x^8 + x^7 is changed to ax + b. Prove that the difference between the integers a and b is always divisible by 49.
Define D to be the differention operator, i.e.
.
Let's say a polynomial p(x) of degree n is "acceptable" if
where is divisible by 49. It is easy to
verify that is acceptable.
It is an immediate consequence that if p is acceptable, then
so is . If we can show that
is acceptable whenever p is acceptable, we will be done.
So suppose p is an acceptable polynomial of degree n; i.e.
where is divisible
by 49. By the extended rule for differentiation of a product
(which has a binomial-theorem-like look to it),
and , which is divisible
by 49; so is acceptable. We're done.