The factorial representation for a polynomial is a way to write a polynomial as a sum of terms, where each term is a product of a coefficient and a factorial expression involving variables.
Suppose we have a polynomial of degree n:
f(x) = a0 + a1x + a2x^2 + ... + anxn
The factorial representation of f(x) is:
f(x) = a0 + a1x + a2x(x - 1) + a3x(x - 1)(x - 2) + ... + anx(x - 1)(x - 2)...(x - n + 1)
In each term of the sum, the coefficient ai is multiplied by a factorial expression that involves the variable x and some lower degree factors. The factorial expression is formed by multiplying together a sequence of decreasing factors, each of which involves x and an integer constant. The first factor is x, the second factor is (x - 1), the third factor is (x - 2), and so on, until the final factor is (x - n + 1).
For example, the factorial representation of a cubic polynomial:
f(x) = 3x^3 - 4x^2 + 5x - 6
is:
f(x) = 3 + 2x + x(x - 1) - x(x - 1)(x - 2)
Note that the first two terms have only constant or linear factorial expressions, while the last two terms have quadratic and cubic factorial expressions. The polynomial is expressed as a sum of terms, where each term has a coefficient multiplied by a factorial expression involving x.