Okay, so I need to find the value of p(8) where p(x) is a polynomial of degree 5. The given information is that p(n) equals n divided by (n² - 1) for n = 2, 3, 4, 5, 6, 7. Since p(x) is a degree 5 polynomial, and we have six points given (from n=2 to n=7), that makes sense because a degree 5 polynomial is uniquely determined by six points. So, the problem is essentially interpolation here.

First, let me write down what I know. The polynomial p(x) satisfies p(n) = n/(n² - 1) for n from 2 to 7. So, at each integer n in that range, the polynomial takes that value. Now, I need to find p(8). Hmm.

I remember that when dealing with polynomial interpolation, especially with given points, sometimes it's helpful to consider constructing a polynomial that has roots at certain points. Maybe there's a way to relate p(x) to another polynomial that has known roots or properties.

Wait, let me think. The given values of n are from 2 to 7, inclusive. The denominator in each case is n² - 1, which factors as (n - 1)(n + 1). So, for each n, the denominator is (n - 1)(n + 1). That means that the denominator is zero when n = 1 or n = -1. But since our points start at n=2, those denominators are non-zero. 

But maybe I can consider the function q(n) = (n² - 1)p(n) - n. If I do that, then for n = 2, 3, ..., 7, q(n) would be zero because (n² - 1)p(n) - n = (n² - 1)(n/(n² - 1)) - n = n - n = 0. So, q(n) has roots at n = 2, 3, 4, 5, 6, 7. Therefore, q(x) is a polynomial that has these six roots. 

Now, the degree of q(x) would be... Let's see. p(x) is degree 5, so (n² - 1)p(n) would be degree 5 + 2 = 7. Then subtracting n (degree 1) would still leave the degree as 7. However, q(x) has roots at six points, so q(x) must be divisible by the polynomial (x - 2)(x - 3)...(x - 7). That polynomial is degree 6. Therefore, q(x) must be equal to (x - 2)(x - 3)...(x - 7) multiplied by some linear factor, since q(x) is degree 7.

So, let's write that as q(x) = (x - 2)(x - 3)(x - 4)(x - 5)(x - 6)(x - 7)(ax + b), where ax + b is a linear polynomial. Then, we can expand this and find the coefficients a and b by using the fact that q(x) = (x² - 1)p(x) - x. But wait, maybe there's another way. Since we know q(x) is equal to both expressions, maybe we can find the leading coefficient or some specific value to solve for a and b.

Alternatively, since q(x) is defined as (x² - 1)p(x) - x, the leading term of q(x) would be the leading term of (x² - 1)p(x). Since p(x) is degree 5, the leading term is, say, c x^5. Then (x² - 1)p(x) would have a leading term c x^7, and subtracting x (degree 1) would leave the leading term as c x^7. On the other hand, in the expression q(x) = (x - 2)...(x - 7)(ax + b), the leading term is a x^7 (since the product of six linear terms gives x^6, multiplied by ax + b gives a x^7 + ...). Therefore, the coefficient a must equal c. But we don't know c yet.

Alternatively, maybe we can consider evaluating q(x) at another point, say x = -1, since the denominator in the original function is (x² - 1), which is zero at x = 1 and x = -1. Let's see. Let me plug x = -1 into the equation q(x) = (x² - 1)p(x) - x. Then, q(-1) = ((-1)^2 - 1)p(-1) - (-1) = (1 - 1)p(-1) + 1 = 0 * p(-1) + 1 = 1. But also, q(-1) is equal to (-1 - 2)(-1 - 3)...(-1 - 7)(a*(-1) + b). Let's compute that product.

First, let's compute the product (-1 - 2)(-1 - 3)...(-1 - 7). That's (-3)(-4)(-5)(-6)(-7)(-8). Let's compute this step by step:

-3 * -4 = 12

12 * -5 = -60

-60 * -6 = 360

360 * -7 = -2520

-2520 * -8 = 20160

So, the product is 20160. Then, multiplying by (a*(-1) + b) gives 20160*(-a + b). But we also know that q(-1) = 1. Therefore:

20160*(-a + b) = 1

So, -a + b = 1 / 20160. Let's note that equation as equation (1).

Similarly, maybe we can evaluate q(1). Let's check x = 1. Then q(1) = (1 - 1)p(1) - 1 = 0 * p(1) - 1 = -1. On the other hand, q(1) is equal to (1 - 2)(1 - 3)...(1 - 7)(a*1 + b). Let's compute that product.

(1 - 2) = -1

(1 - 3) = -2

(1 - 4) = -3

(1 - 5) = -4

(1 - 6) = -5

(1 - 7) = -6

So, the product is (-1)(-2)(-3)(-4)(-5)(-6). Let's compute this:

(-1)*(-2) = 2

2*(-3) = -6

-6*(-4) = 24

24*(-5) = -120

-120*(-6) = 720

So, the product is 720. Then, multiplying by (a*1 + b) gives 720*(a + b). But q(1) = -1, so:

720*(a + b) = -1

Therefore, a + b = -1 / 720. Let's note that as equation (2).

Now, we have two equations:

(1) -a + b = 1 / 20160

(2) a + b = -1 / 720

We can solve this system of equations for a and b. Let's add the two equations:

(-a + b) + (a + b) = (1 / 20160) + (-1 / 720)

Simplify left side: 2b

Right side: 1/20160 - 1/720. Let's convert 1/720 to the same denominator as 20160. 720 * 28 = 20160, so 1/720 = 28/20160.

Therefore, 1/20160 - 28/20160 = (1 - 28)/20160 = (-27)/20160 = -9/6720 = -3/2240. Wait, let me check that again.

Wait, 1/720 is equal to 28/20160 because 720 * 28 = 20160. So, 1/720 = 28/20160. Therefore, 1/20160 - 28/20160 = -27/20160. Simplify that fraction:

Divide numerator and denominator by 3: -9/6720. Then divide numerator and denominator by 3 again: -3/2240. So, 2b = -3/2240. Therefore, b = -3/(2*2240) = -3/4480.

Then, plug b into equation (2): a + (-3/4480) = -1/720. Therefore, a = -1/720 + 3/4480. Let's compute that.

First, find a common denominator for 720 and 4480. Let's see, 720 factors into 2^4 * 3^2 * 5, and 4480 is 448 * 10 = 64 * 7 * 2 * 5 = 2^6 * 5 * 7. So, the least common multiple would be 2^6 * 3^2 * 5 * 7 = 64 * 9 * 5 * 7 = 64 * 315 = 20160. So, convert both fractions to denominator 20160.

-1/720 = -28/20160

3/4480 = (3 * 4.5)/20160 = 13.5/20160. Wait, but we can't have fractions here. Let's compute 3/4480 in 20160 denominator:

20160 / 4480 = 4.5. Hmm, that's a decimal. Maybe better to compute numerators:

-1/720 = -28/20160

3/4480 = (3 * 4.5)/20160 = 13.5/20160. But since we can't have fractions in numerator, maybe we should use exact fractions.

Alternatively, let's compute -1/720 + 3/4480:

Convert to decimal:

-1/720 ≈ -0.001388888...

3/4480 ≈ 0.000669642857...

Adding them: approximately -0.000719245...

But maybe exact fraction:

-1/720 + 3/4480 = (-4480 + 3*720) / (720*4480)

Wait, no. Let me compute it properly.

To add the fractions, find a common denominator, which is 20160.

-1/720 = -28/20160

3/4480 = (3 * 4.5)/20160 = 13.5/20160. But since 4480 * 4.5 = 20160. However, fractions with denominators not dividing into integers can be tricky. Maybe instead, multiply numerator and denominator by appropriate factors.

Alternatively, compute:

-1/720 + 3/4480 = (-1 * 4480 + 3 * 720) / (720 * 4480)

Calculate numerator: -4480 + 2160 = -2320

Denominator: 720 * 4480 = 3,225,600

Therefore, a = -2320 / 3,225,600. Simplify this fraction.

Divide numerator and denominator by 80: -2320 ÷ 80 = -29; 3,225,600 ÷ 80 = 40,320.

So, -29/40,320. Check if this can be simplified. 29 is a prime number. 40,320 ÷ 29 = approximately 1390.34, which is not an integer. So, the simplified fraction is -29/40,320.

Therefore, a = -29/40,320 and b = -3/4,480. Wait, but 4,480 is 4480. Let me confirm:

Wait, 40,320 is 20160 * 2, right? 20160 * 2 = 40,320. And 4480 is 20160 / 4.5, which isn't an integer. Maybe I made a mistake in the calculation.

Wait, let's check the previous step again. When adding -1/720 + 3/4480:

First, convert both fractions to have denominator 20160:

-1/720 = -28/20160

3/4480 = (3 * 4.5)/20160 = 13.5/20160. But 13.5 is not an integer. So, perhaps instead, use the least common multiple (LCM) of 720 and 4480, which is 20160. So, 3/4480 = (3 * 4.5)/20160. Hmm, but 4.5 is not an integer. Maybe this approach isn't working. Let me instead convert both fractions to decimal to check:

-1/720 ≈ -0.001388888...

3/4480 ≈ 0.000669642857...

Adding them: -0.001388888... + 0.000669642857... ≈ -0.000719245...

So, approximately -0.000719245. Let's see if this is equal to -29/40,320.

Calculate 29/40,320 ≈ 0.000719245... Yes, that's correct. So, -29/40,320 ≈ -0.000719245, which matches the decimal. Therefore, a = -29/40,320.

So, putting it all together, q(x) = (x - 2)(x - 3)...(x - 7)(ax + b) where a = -29/40,320 and b = -3/4,480. Wait, but earlier we had b = -3/4480, which is -3/4480. Let me confirm that.

From equation (1): -a + b = 1/20160

From equation (2): a + b = -1/720

We found a = -29/40,320 and b = -3/4480. Let's check if these satisfy both equations.

First equation:

-a + b = 29/40,320 + (-3/4480) = 29/40,320 - 3/4480. Convert 3/4480 to denominator 40,320: 40,320 / 4480 = 9, so 3/4480 = 27/40,320. Therefore, 29/40,320 - 27/40,320 = 2/40,320 = 1/20,160. Which matches equation (1). Good.

Second equation:

a + b = (-29/40,320) + (-3/4480) = -29/40,320 - 27/40,320 = -56/40,320 = -1/720. Which matches equation (2). Perfect.

So, now we have a and b. Therefore, the expression for q(x) is (x - 2)(x - 3)...(x - 7)(-29/40,320 x - 3/4480). Then, since q(x) = (x² - 1)p(x) - x, we can solve for p(x):

(x² - 1)p(x) - x = (x - 2)(x - 3)...(x - 7)(-29/40,320 x - 3/4480)

Therefore,

(x² - 1)p(x) = x + (x - 2)(x - 3)...(x - 7)(-29/40,320 x - 3/4480)

Then,

p(x) = [x + (x - 2)(x - 3)...(x - 7)(-29/40,320 x - 3/4480)] / (x² - 1)

But we need to compute p(8). So, let's substitute x = 8 into this expression.

First, compute the denominator: 8² - 1 = 64 - 1 = 63.

Now, compute the numerator: 8 + (8 - 2)(8 - 3)...(8 - 7)(-29/40,320 * 8 - 3/4480)

First, compute the product (8 - 2)(8 - 3)(8 - 4)(8 - 5)(8 - 6)(8 - 7). That's 6*5*4*3*2*1 = 720.

Then, compute the linear term: (-29/40,320)*8 - 3/4480.

Compute each part:

First term: (-29/40,320)*8 = (-29*8)/40,320 = (-232)/40,320. Simplify: divide numerator and denominator by 8: -29/5,040.

Second term: -3/4480. Let's convert both fractions to have denominator 5,040* something. Wait, 40,320 and 4480. Let's find a common denominator for -29/5,040 and -3/4,480.

Wait, maybe compute decimal equivalents:

-29/5,040 ≈ -0.005755...

-3/4,480 ≈ -0.00066964...

Adding them: -0.005755 - 0.00066964 ≈ -0.0064246...

But let's compute exactly:

Convert -29/5,040 and -3/4,480 to a common denominator. The denominators are 5,040 and 4,480. Let's compute LCM of 5,040 and 4,480. Since 5,040 = 7*720, 4,480 = 6.2857*720. Wait, better to factor:

5,040 = 2^4 * 3^2 * 5 * 7

4,480 = 2^6 * 5 * 7

So, LCM is 2^6 * 3^2 * 5 * 7 = 64 * 9 * 5 * 7 = 64 * 315 = 20,160.

Therefore, convert both fractions to denominator 20,160.

-29/5,040 = (-29 * 4)/20,160 = -116/20,160

-3/4,480 = (-3 * 4.5)/20,160 = -13.5/20,160. But again, fractional numerator. Alternatively, perhaps better to use decimal.

Alternatively, note that -29/5,040 - 3/4,480 = (-29*4.5 - 3*5.7)/20,160. Wait, 5,040 * 4.5 = 22,680, which is not 20,160. Hmm, maybe this isn't the right approach.

Alternatively, compute the exact decimal:

First term: -29/5,040 ≈ -0.005755555...

Second term: -3/4,480 ≈ -0.000669642857...

Adding them: -0.005755555... - 0.000669642857... ≈ -0.006425198...

But let's compute this precisely:

-29/5,040 - 3/4,480 = (-29*4.5 - 3*5.7)/20,160. Wait, 20,160 is LCM. Wait, 5,040 * 4 = 20,160, and 4,480 * 4.5 = 20,160. So, multiply numerator and denominator accordingly.

First term: (-29/5,040) = (-29 * 4)/20,160 = -116/20,160

Second term: (-3/4,480) = (-3 * 4.5)/20,160 = -13.5/20,160

Therefore, total is (-116 - 13.5)/20,160 = -129.5/20,160. Which is equal to -259/40,320. Because 129.5 * 2 = 259, and 20,160 * 2 = 40,320. Therefore, the linear term is -259/40,320.

Therefore, the linear term is -259/40,320.

Therefore, the entire numerator becomes:

8 + (720)(-259/40,320) = 8 - (720 * 259)/40,320.

Compute (720 * 259)/40,320:

First, 720 and 40,320 have a common factor. 40,320 ÷ 720 = 56. So, 720 * 259 / 40,320 = 259 / 56.

Therefore, the numerator is 8 - 259/56. Convert 8 to 56ths: 8 = 448/56. So, 448/56 - 259/56 = (448 - 259)/56 = 189/56.

Therefore, numerator is 189/56.

Therefore, p(8) = (189/56) / 63 = (189/56) * (1/63) = 189 / (56*63) = 189 / 3528.

Simplify 189/3528. Let's divide numerator and denominator by 21: 189 ÷ 21 = 9; 3528 ÷ 21 = 168. So, 9/168. Then divide numerator and denominator by 3: 3/56. So, p(8) = 3/56.

Wait, let me check that again.

Wait, 189 divided by 3528. Let's see:

Divide numerator and denominator by 21: 189 ÷21=9, 3528 ÷21=168. So, 9/168. Then divide numerator and denominator by 3: 3/56. Yes, that's correct. So, 3/56.

Therefore, the final answer is 3/56. Let me just verify the steps again to make sure I didn't make a mistake.

First, q(x) = (x² - 1)p(x) - x. For x = n from 2 to 7, q(n) = 0. So, q(x) has roots at 2 to 7, so q(x) = (x - 2)...(x - 7)(ax + b). Then, we found q(-1) and q(1) to get equations for a and b. Solved and got a = -29/40,320 and b = -3/4480. Then, substituted x =8 into q(x) to find the numerator for p(8). The numerator was 8 + product*(linear term). Product was 720, linear term was -259/40,320. Then, 720*(-259)/40,320 = -259/56. So, 8 - 259/56 = 8 - 4.625 = 3.375, which is 27/8. Wait, wait a second, this contradicts the previous calculation.

Wait, hold on. Let me recheck the numerator calculation.

Wait, the numerator is 8 + (720)(-259/40,320). Let's compute 720 * (-259)/40,320.

720 / 40,320 = 1/56. So, 720 * (-259)/40,320 = (-259)/56. Therefore, the numerator is 8 + (-259/56). Now, 8 is 448/56. So, 448/56 - 259/56 = (448 - 259)/56 = 189/56. Then, p(8) = numerator / denominator = (189/56)/63 = 189/(56*63) = 189/3528 = 3/56. So, that's correct.

But when I thought about 8 - 259/56, 259/56 is approximately 4.625, so 8 - 4.625 is 3.375, which is 27/8. Wait, but 189/56 is approximately 3.375, which is 27/8. Wait, 189 divided by 56: 56*3=168, 189-168=21, so 3 + 21/56 = 3 + 3/8 = 3.375. Yes, 189/56 is 3.375, which is 27/8. Wait, 27/8 is 3.375 as well. Wait, but 189/56 simplifies to 27/8? Let's check:

189 ÷ 7 = 27, 56 ÷ 7 = 8. So, 189/56 = 27/8. Yes! Therefore, numerator is 27/8, and denominator is 63. So, p(8) = (27/8)/63 = 27/(8*63) = 27/504 = 3/56. So, same answer. So, that's correct.

Therefore, the answer is 3/56. So, I think that's the correct result.

But let me cross-verify using another method. Maybe using Lagrange interpolation.

Given that p(n) = n/(n² -1) for n=2,3,...,7, and we need p(8). Since p(x) is a degree 5 polynomial, and we have 6 points, we can use Lagrange's formula.

But constructing the Lagrange polynomial for 6 points is quite tedious, but maybe we can use the concept of constructing p(x) as a sum of basis polynomials each corresponding to a point.

Alternatively, since the problem is similar to constructing p(x) such that (x² -1)p(x) -x has roots at 2,3,...,7, which we already did. So, since that approach worked, and the answer came out as 3/56, which seems reasonable.

Alternatively, let's test with n=8. Let's compute p(8) using the expression we found. Let's see if 3/56 is correct.

Alternatively, let's see if the answer makes sense. For n=2, p(2)=2/(4-1)=2/3≈0.6667. Then, as n increases, n/(n² -1) decreases. For example, at n=3, it's 3/8=0.375, n=4, 4/15≈0.2667, n=5, 5/24≈0.2083, n=6, 6/35≈0.1714, n=7, 7/48≈0.1458. So, these values are decreasing. So, p(8) should be less than 7/48≈0.1458. 3/56≈0.0536, which is indeed less. So, that seems plausible.

Alternatively, think of the function x/(x² -1). For x=8, that's 8/(64-1)=8/63≈0.12698. So, p(8) is 3/56≈0.05357, which is less than 8/(8² -1)=8/63. So, that seems reasonable, as p(x) is a degree 5 polynomial that interpolates x/(x² -1) at x=2 to 7, so at x=8, it's not necessarily equal to x/(x² -1). So, the answer seems plausible.

Therefore, after careful calculation and verification, I believe the answer is 3/56.
Therefore, the final answer is $\boxed{\frac{3}{56}}$.
</think>