1. ## Polynomial functions - finite difference - leading coefficient

Can someone explain to me how you can find out what the leading coefficient is from a finite difference.

For example, the function
ax^5 + 3x^4 - 2x^3 - 3x^2 + x - 1
has a common difference of -120.
What is the value of a?

You don't have to tell me the answer of this question, just the process of how to find it. I want to be able to figure this out myself, but I have no clue where to start. Any hints or leads are GREATLY APPRECIATED!!!

Thanks!

2. First, you need to make more sense. How can a polynomial have a common difference? What does that mean?

On the other hand, your answer may be hiding in successive derivatives.

If p(x) = ax^5 + 3x^4 - 2x^3 - 3x^2 + x - 1

Then p'(x) = 5ax^4 + 12x^3 - 6x^2 - 6x^1 + 1

Then p''(x) = 20ax^3 + 36x^2 - 12x^1 - 6

Then p'''(x) = 60ax^2 + 72x^1 - 12

Then p''''(x) = 120ax^1 + 72

Then p'''''(x) = 120a = 120

3. First, you need to make more sense. How can a polynomial have a common difference? What does that mean?
Don't tell me, tell my text book! That term was copied directly from the question in the text, so it's not my fault if it doesn't make sense! I imagine they mean "finite difference" since that is what we have been dealing with thus far.

I have been told via my text that you can use finite differences to determine the leading coefficient of any polynomial function.

But thanks for the process you offered here, I will try to apply it to my other functions.

4. The idea is learning, not blaming. If the book doesn't make sense, and you quote the book, you don't make sense either.

You need VALUES for finite differences.

Produce some VALUES.
Calculate up to 5th Differences.
Look at the common value.

p(0) = -1
p(1) = a - 2
p(-1) = -a

Do 3 more and start calculating differences.

5. ## Middle school math?

I doubt they're teaching this in middle schools today.

6. From the looks of it, the very poorly worded question, they shouldn't be.

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