Let $ x_1, x_2, x_3, x_4 $ be the solution of the equation $$ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 $$ and $$ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m.$$ Then the value of $ m $ is ______.

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We are given a polynomial equation $ P(x) = 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 $ with roots $ x_1, x_2, x_3, x_4 $. We need to find the value of $ m $ from the given relation $ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m $. Concept Used: If a polynomial $ P(x) $ of degree $ n $ has a leading coefficient $ a_n $ and roots $ x_1, x_2, \ldots, x_n $, it can be written in factored form as: $$ P(x) = a_n (x - x_1)(x - x_2) \cdots (x - x_n) $$ The product $ \prod_{i=1}^{n} (k^2 + x_i^2) $ can be evaluated by considering the polynomial at complex values. Specifically, we use the identity $ k^2 + x_i^2 = -( -k^2 - x_i^2 ) = -( (ik)^2 - x_i^2 ) = (x_i - ik)(x_i + ik) $. This suggests evaluating $ P(ik) $ and $ P(-ik) $. Since the coefficients of $ P(x) $ are real, we have $ P(\bar{z}) = \overline{P(z)} $. Therefore, $ P(-ik) = \overline{P(ik)} $, and their product is $ P(ik)P(-ik) = |P(ik)|^2 $. Step-by-Step Solution: Step 1: Write the polynomial in its factored form. The given polynomial is $ P(x) = 4x^4 + 8x^3 - 17x^2 - 12x + 9 $. Since its roots are $ x_1, x_2, x_3, x_4 $, we can write: $$ P(x) = 4(x - x_1)(x - x_2)(x - x_3)(x - x_4) $$ Step 2: Evaluate the polynomial at $ x = 2i $ and $ x = -2i $. Substituting $ x = 2i $: $$ P(2i) = 4(2i - x_1)(2i - x_2)(2i - x_3)(2i - x_4) $$ Substituting $ x = -2i $: $$ P(-2i) = 4(-2i - x_1)(-2i - x_2)(-2i - x_3)(-2i - x_4) $$ Step 3: Multiply $ P(2i) $ and $ P(-2i) $ to form the desired expression. $$ P(2i) P(-2i) = 16 \prod_{k=1}^{4} (2i - x_k)(-2i - x_k) $$ For each term in the product, we have: $$ (2i - x_k)(-2i - x_k) = (-x_k + 2i)(-x_k - 2i) = (-x_k)^2 - (2i)^2 = x_k^2 - (-4) = x_k^2 + 4 $$ Therefore, the product becomes: $$ P(2i) P(-2i) = 16 \left(x_1^2 + 4\right)\left(x_2^2 + 4\right)\left(x_3^2 + 4\right)\left(x_4^2 + 4\right) $$ From this, we can express the desired quantity as: $$ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{P(2i)P(-2i)}{16} $$ Step 4: Calculate the value of $ P(2i) $. $$ P(2i) = 4(2i)^4 + 8(2i)^3 - 17(2i)^2 - 12(2i) + 9 $$ We use the powers of $ i $: $ i^2 = -1, i^3 = -i, i^4 = 1 $. $$ P(2i) = 4(16 i^4) + 8(8 i^3) - 17(4 i^2) - 24i + 9 $$ $$ P(2i) = 4(16) + 8(-8i) - 17(-4) - 24i + 9 $$ $$ P(2i) = 64 - 64i + 68 - 24i + 9 $$ Combining the real and imaginary parts: $$ P(2i) = (64 + 68 + 9) + (-64 - 24)i = 141 - 88i $$ Step 5: Calculate $ P(2i)P(-2i) $. Since the coefficients of $ P(x) $ are real, $ P(-2i) = \overline{P(2i)} = 141 + 88i $. Thus, $ P(2i)P(-2i) = |P(2i)|^2 $. $$ |P(2i)|^2 = (141)^2 + (-88)^2 $$ $$ (141)^2 = 19881 $$ $$ (-88)^2 = 7744 $$ $$ |P(2i)|^2 = 19881 + 7744 = 27625 $$ Final Computation & Result: Step 6: Substitute this value back into the expression from Step 3. $$ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{27625}{16} $$ Step 7: Equate this result to the given expression to find $ m $. $$ \frac{27625}{16} = \frac{125}{16} m $$ Canceling $ \frac{1}{16} $ from both sides, we get: $$ 27625 = 125 m $$ Solving for $ m $: $$ m = \frac{27625}{125} $$ $$ m = 221 $$ Therefore, the value of $ m $ is 221 .

Let \( x_1, x_2, x_3, x_4 \) be the solution of the equation \[ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 \] and \[ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m.\] Then the value of \( m \) is ______.

Correct Answer: 221

Approach Solution - 1

Concept Used:

Step-by-Step Solution:

Final Computation & Result:

Approach Solution -2

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