Okay, let me try to figure out this problem. So, we have a quadratic polynomial p(x) with integer coefficients, and one of its roots is 4 - sqrt(11). We need to compute p(3)/p(4). Hmm, quadratics with integer coefficients... right, if a polynomial has integer coefficients and a root that's irrational, like 4 - sqrt(11), then the other root must be its conjugate, right? Because irrational roots come in conjugate pairs when the coefficients are rational. Since the coefficients here are integers, which are a subset of rational numbers, the same principle applies. So, the other root should be 4 + sqrt(11). Let me confirm that. If the polynomial has integer coefficients, then when you have a root like a + b√c, the conjugate a - b√c must also be a root. Yeah, that makes sense because when you multiply the factors (x - (a + b√c))(x - (a - b√c)), the cross terms involving √c will cancel out, leaving integer coefficients. So, the two roots are 4 - sqrt(11) and 4 + sqrt(11).

Therefore, the quadratic polynomial can be written as p(x) = k(x - (4 - sqrt(11)))(x - (4 + sqrt(11))), where k is an integer. Since the leading coefficient is an integer, k must be an integer. But since we're dealing with ratios p(3)/p(4), maybe the constant k will cancel out. Let me check that. If p(x) is k times the product, then p(3) would be k*(3 - (4 - sqrt(11)))*(3 - (4 + sqrt(11))) and p(4) would be k*(4 - (4 - sqrt(11)))*(4 - (4 + sqrt(11))). Then when we take the ratio p(3)/p(4), the k cancels, so we only need to compute the product of those terms divided by each other. So, the value of k doesn't matter here. That's good to know. So, we can proceed without worrying about the value of k.

Let me compute p(3) first. Let's expand the product first. Let's write the quadratic polynomial in standard form. Let's compute (x - (4 - sqrt(11)))(x - (4 + sqrt(11))). Using the difference of squares formula: (x - 4 + sqrt(11))(x - 4 - sqrt(11)) = (x - 4)^2 - (sqrt(11))^2. Let's compute that. (x - 4)^2 is x^2 - 8x + 16, and (sqrt(11))^2 is 11. So, subtracting those gives x^2 - 8x + 16 - 11 = x^2 - 8x + 5. Therefore, the quadratic polynomial is p(x) = k(x^2 - 8x + 5). So, the standard form is x squared minus 8x plus 5, multiplied by an integer k. But since we need to compute the ratio p(3)/p(4), the k cancels, so we can ignore it. Therefore, p(3) = (3)^2 - 8*(3) + 5, and p(4) = (4)^2 - 8*(4) + 5. Let me compute these.

First, p(3): 3 squared is 9, minus 8*3 is 24, plus 5. So, 9 - 24 + 5. 9 + 5 is 14, minus 24 is -10. So, p(3) is -10. Then p(4): 4 squared is 16, minus 8*4 is 32, plus 5. So, 16 - 32 + 5. 16 + 5 is 21, minus 32 is -11. So, p(4) is -11. Therefore, the ratio p(3)/p(4) is (-10)/(-11) which simplifies to 10/11. So, the answer should be 10/11. Hmm, that seems straightforward. But let me verify my steps again to make sure I didn't make a mistake.

First, confirming the roots: if one root is 4 - sqrt(11), the conjugate is 4 + sqrt(11). So, the quadratic is (x - (4 - sqrt(11)))(x - (4 + sqrt(11))) = x^2 - 8x + 5. Correct. Then p(3) is 3^2 -8*3 +5 = 9 -24 +5 = -10. p(4) is 16 -32 +5 = -11. So, the ratio is 10/11. The negative signs cancel, giving a positive 10/11. That seems right. So, the answer is 10/11.

But wait, let me think again. The problem says "quadratic polynomial with integer coefficients which has 4 - sqrt(11) as a root". So, the minimal polynomial with integer coefficients would indeed be x^2 -8x +5. If there's a different quadratic with integer coefficients having that root, it would be a multiple of this polynomial. But since the problem doesn't specify any leading coefficient, we can take k=1, and that would be the minimal case. But since the ratio p(3)/p(4) is the same regardless of k, as we saw earlier, because k cancels. So, even if the polynomial was 2x^2 -16x +10, then p(3) would be 2*(9) -16*3 +10 = 18 -48 +10 = -20, and p(4) would be 2*16 -16*4 +10 =32 -64 +10 =-22, so ratio is (-20)/(-22)=10/11. Same result. So, regardless of the leading coefficient k, the ratio is 10/11. Therefore, the answer is 10/11.

Wait, so maybe there's an alternative way to approach this problem? For example, using Vieta's formula. Let's see. Vieta's formula relates the sum and product of roots. For a quadratic equation ax^2 + bx + c =0, the sum of roots is -b/a, and the product is c/a. In our case, the sum of the roots is (4 - sqrt(11)) + (4 + sqrt(11)) = 8. The product is (4 - sqrt(11))(4 + sqrt(11)) = 16 -11=5. So, the quadratic can be written as x^2 - (sum)x + product = x^2 -8x +5. So, same as before. Then evaluating p(3) and p(4) gives us -10 and -11, ratio 10/11. So, same result. So, confirms the answer.

Alternatively, if I didn't remember Vieta's formulas, I could just compute the product directly. So, p(3) = (3 - (4 - sqrt(11)))(3 - (4 + sqrt(11))) = (3 -4 + sqrt(11))(3 -4 - sqrt(11)) = (-1 + sqrt(11))*(-1 - sqrt(11)) = (-1)^2 - (sqrt(11))^2 = 1 -11 = -10. Similarly, p(4) = (4 - (4 - sqrt(11)))(4 - (4 + sqrt(11))) = (sqrt(11))*(-sqrt(11)) = -11. So, same result. So, ratio is 10/11. So, that's another way to compute p(3) and p(4) directly without expanding the quadratic. It's interesting. So, even if we didn't know the quadratic, by directly computing the product at x=3 and x=4, we can get the same answer. That's a neat trick.

So, in summary, since the quadratic must have the conjugate root, we can construct the polynomial from the two roots, then evaluate at 3 and 4. The ratio simplifies to 10/11. Therefore, the answer is 10/11. I think that's solid. Let me check if there's any possible mistake here. For instance, if I miscalculated the product when evaluating p(3). Let's recompute p(3):

(3 - (4 - sqrt(11))) = 3 -4 + sqrt(11) = -1 + sqrt(11)

(3 - (4 + sqrt(11))) = 3 -4 - sqrt(11) = -1 - sqrt(11)

Multiply these two: (-1 + sqrt(11))*(-1 - sqrt(11)) = (-1)^2 - (sqrt(11))^2 = 1 -11 = -10. Correct.

Similarly, p(4):

(4 - (4 - sqrt(11))) = sqrt(11)

(4 - (4 + sqrt(11))) = -sqrt(11)

Multiply them: sqrt(11)*(-sqrt(11)) = -11. Correct.

Therefore, ratio is -10/-11 = 10/11. So, yes, that's correct. So, I think this is the right answer. I don't see any errors in the reasoning. All the steps check out. So, the final answer is 10/11.

**Final Answer**
The value of $\frac{p(3)}{p(4)}$ is $\boxed{\frac{10}{11}}$.
Therefore, $\frac{p(3)}{p(4)} = \frac{10}{11}$.
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