If f(x) is poly w/ degree d, show f(x)-f(x-1) is poly of degree d-1.

[nmego12345]

Question is below:

a. If f (x) is a non-zero polynomial of degree d, show that:

. . . . .\(\displaystyle \left(\Delta\, f\right)(x)\, =\, f(x)\, -\, f(x\, -\, 1)\)

is a polynomial of degree d - 1. (For the purposes of this result, it is convenient to define the degree of the zero polynomial to be -1.)

b. If f (x) = g (x) + h (x), show that:

. . . . .\(\displaystyle \Delta\, f\, =\, \Delta\, g\, +\, \Delta\, h\)

c. Use the binomial theorem to calculate:

. . . . .\(\displaystyle \Delta\, \dfrac{x^{k+1}}{k\, +\, 1}\)

d. Suppose that, for all i < k, one knows a polynomial f_i, such that:

. . . . .\(\displaystyle \left(\Delta\, f_i\right)(x)\, =\, x^i\)

The last problem gave us f_i for i = 0, 1, 2, 3, and 4. Combine your work on this problem in order to show you how to calculate f_k+1.

So far, I have solved parts a, b, and c. I don't know how to solve d.

 

[stapel]

a. If f (x) is a non-zero polynomial of degree d, show that:

. . . . .\(\displaystyle \left(\Delta\, f\right)(x)\, =\, f(x)\, -\, f(x\, -\, 1)\)

is a polynomial of degree d - 1. (For the purposes of this result, it is convenient to define the degree of the zero polynomial to be -1.)

b. If f (x) = g (x) + h (x), show that:

. . . . .\(\displaystyle \Delta\, f\, =\, \Delta\, g\, +\, \Delta\, h\)

c. Use the binomial theorem to calculate:

. . . . .\(\displaystyle \Delta\, \dfrac{x^{k+1}}{k\, +\, 1}\)

d. Suppose that, for all i < k, one knows a polynomial f_i, such that:

. . . . .\(\displaystyle \left(\Delta\, f_i\right)(x)\, =\, x^i\)

The last problem gave us f_i for i = 0, 1, 2, 3, and 4. Combine your work on this problem in order to show you how to calculate f_k+1.
So far, I have solved parts a, b, and c. I don't know how to solve d.

What was "the last problem"? Thank you!