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
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.
Approach Solution -2
The given polynomial can be expressed as:
\[ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 4(x - x_1)(x - x_2)(x - x_3)(x - x_4). \]
Let \(x_1 = 2i\) and \(x_2 = -2i\). Substituting these values:
\[ 64 - 64i + 68 - 24i + 9 = 4(2i - x_1)(2i - x_2)(2i - x_3)(2i - x_4). \]
Simplify:
\[ 141 - 88i \quad \dots \quad (1) \]
Similarly, for \(-2i\):
\[ 64 + 64i + 68 + 24i + 9 = 4(-2i - x_1)(-2i - x_2)(-2i - x_3)(-2i - x_4). \]
Simplify:
\[ 141 + 88i \quad \dots \quad (2) \]
Using the given condition:
\[ \frac{125}{16}m = \frac{141^2 + 88^2}{16}. \]
Calculate:
\[ m = 221. \]
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