Find the original Polynomial

[Dyl]

There is a polynomial of degree 4. It can be expressed as P(x) = n/(n+1) for 0 <= n <= 4 for n E Z
Find the value of the polynomial if n = 5. (2/3 is the answer)

From what I have if you express the equation as P(x) = ax^4 + bx^3 + cx^2 + dx + e, and you sub P(0) then 0 + e = 0, which means that e is then 0.
So youre left with P(x) = x(ax^3 + bx^2 + cx + d)

P(1) = 1/2 so: (a + b + c + d) = 1/2
P(2) = 2/3 so: 2(8a+4b +2c +d) = 2/3. 8a + 4b + 2c + d = 1/3

 

[Steven G]

Can we please see the whole problem? That really would be helpful!
Why is P(0)=0? After all, if x=0 that does NOT mean that n=0
Why is P(1) = 1/2? After all, if x=1 that does NOT mean that n=1
Why is P(2)=2/3? After all, if x=2 that does NOT mean that n=2
If n=5, then how is 0 <= n <= 4 ?
Also if P(x) = n/(n+1) for 0 <= n <= 4 for n E Z, then P(x) can only take on at most 5 values!
What is the connection between x and n??

If you meant to say p(x) = x/(x+1) = 1 - 1/(x+1) then p(x) is NOT a polynomial of any degree

 

[Dr.Peterson]

There is a polynomial of degree 4. It can be expressed as P(x) = n/(n+1) for 0 <= n <= 4 for n E Z
Find the value of the polynomial if n = 5. (2/3 is the answer)

From what I have if you express the equation as P(x) = ax^4 + bx^3 + cx^2 + dx + e, and you sub P(0) then 0 + e = 0, which means that e is then 0.
So youre left with P(x) = x(ax^3 + bx^2 + cx + d)

P(1) = 1/2 so: (a + b + c + d) = 1/2
P(2) = 2/3 so: 2(8a+4b +2c +d) = 2/3. 8a + 4b + 2c + d = 1/3

As I read this, you meant to say P(n) = n/(n+1) for n = 0, 1, 2, 3, 4. That is, five values are specified, which is enough to determine the polynomial (degree 4):

• P(0) = 0/1
• P(1) = 1/2
• P(2) = 2/3
• P(3) = 3/4
• P(4) = 4/5

Just continue what you are doing to find the polynomial, then evaluate P(5). What is your question?

There may be a shortcut to get P(5) without actually finding the polynomial, but that isn't really necessary.

 

[Dyl]

 

[MarkFL]

If it helps, here is a graph of the quartic polynomial:



You've already correctly determined we must have the form:

[MATH]f(n)=an^4+bn^3+cn^2+dn[/MATH]
I agree with Dr. Peterson in that proceeding as you have begun will allow you to determine the values of the 4 parameters.