Finite difference of polynomial function, and leading coeff.

[huhrosa]

Q1: You can use finite differences to determine the leading coefficient of any polynomial function. Determine the relation that exists between the nth finite difference and the leading coefficient, a, for the general polynomial function.

f(x) = a[sub:3igctqvy]n[/sub:3igctqvy]x[sup:3igctqvy]n[/sup:3igctqvy] + a[sub:3igctqvy]n-1[/sub:3igctqvy]x[sup:3igctqvy]n-1[/sup:3igctqvy] +...+ a[sub:3igctqvy]2[/sub:3igctqvy]x[sup:3igctqvy]2[/sup:3igctqvy] + a[sub:3igctqvy]1[/sub:3igctqvy]x + a0

I don't understand the relationship between those too.. .the answer at the back of the book says "a[sub:3igctqvy]n[/sub:3igctqvy]n!", which i also dont understand

Q2: The polynomial f(x) = ax[sup:3igctqvy]5[/sup:3igctqvy] + 3x[sup:3igctqvy]4[/sup:3igctqvy] - 2x[sup:3igctqvy]3[/sup:3igctqvy] - 3x[sup:3igctqvy]2[/sup:3igctqvy] + x - 1 has a common difference of -120. What is the value of a?

Because i didnt understand the relationship, I couldn't solve the second problem either. the answer at the back of the book states "-1"

I will appreciate your help

Thank you!

 

[stapel]

Q1: You can use finite differences to determine the leading coefficient of any polynomial function. Determine the relation that exists between the nth finite difference and the leading coefficient, a, for the general polynomial function.

f(x) = a[sub:2s824s2f]n[/sub:2s824s2f]x[sup:2s824s2f]n[/sup:2s824s2f] + a[sub:2s824s2f]n-1[/sub:2s824s2f]x[sup:2s824s2f]n-1[/sup:2s824s2f] +...+ a[sub:2s824s2f]2[/sub:2s824s2f]x[sup:2s824s2f]2[/sup:2s824s2f] + a[sub:2s824s2f]1[/sub:2s824s2f]x + a0

Take the finite differences, and see what you get. If you don't "see" what they're getting at, please reply showing the differences you found, and telling us what your thoughts are so far.

Q2: The polynomial f(x) = ax[sup:2s824s2f]5[/sup:2s824s2f] + 3x[sup:2s824s2f]4[/sup:2s824s2f] - 2x[sup:2s824s2f]3[/sup:2s824s2f] - 3x[sup:2s824s2f]2[/sup:2s824s2f] + x - 1 has a common difference of -120. What is the value of a?

You were given a formula in answer to the previous exercise, and you've taken the differences until you arrived at a common difference (except for where the "a" is messing things up) of -120. You've set the "a" terms equal to the given common-difference value, and have also tried plugging the n-value you determined into the given formula, and... then what?

Please reply showing all of your work and reasoning so far. Thank you!

Eliz.