Question

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 ______.

Answer 1

Correct Answer: 221

Given the polynomial equation \( P(x) = 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 \) with roots \( x_1, x_2, x_3, x_4 \), we are tasked with determining the value of \( m \) using the relationship \( \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m \).

Concept Used:

A polynomial \( P(x) \) of degree \( n \) with leading coefficient \( a_n \) and roots \( x_1, x_2, \ldots, x_n \) can be expressed in factored form as \( P(x) = a_n \prod_{i=1}^{n} (x - x_i) \). The product \( \prod_{i=1}^{n} (k^2 + x_i^2) \) can be computed by evaluating the polynomial at complex numbers. Specifically, using the identity \( k^2 + x_i^2 = (x_i - ik)(x_i + ik) \), we can evaluate \( P(ik) \) and \( P(-ik) \). For polynomials with real coefficients, \( P(\bar{z}) = \overline{P(z)} \), implying \( P(-ik) = \overline{P(ik)} \) and their product is \( P(ik)P(-ik) = |P(ik)|^2 \).

Step-by-Step Solution:

Step 1: Express the polynomial in its factored form.

For the given polynomial \( P(x) = 4x^4 + 8x^3 - 17x^2 - 12x + 9 \) with roots \( x_1, x_2, x_3, x_4 \), the factored form is:

\[P(x) = 4(x - x_1)(x - x_2)(x - x_3)(x - x_4)\]

Step 2: Evaluate the polynomial at \( x = 2i \) and \( x = -2i \).

Substituting \( x = 2i \) yields:

\[P(2i) = 4(2i - x_1)(2i - x_2)(2i - x_3)(2i - x_4)\]

Substituting \( x = -2i \) yields:

\[P(-2i) = 4(-2i - x_1)(-2i - x_2)(-2i - x_3)(-2i - x_4)\]

Step 3: Calculate the product \( P(2i) P(-2i) \).

\[P(2i) P(-2i) = 16 \prod_{k=1}^{4} (2i - x_k)(-2i - x_k)\]

Each term in the product simplifies as:

\[(2i - x_k)(-2i - x_k) = (-x_k + 2i)(-x_k - 2i) = (-x_k)^2 - (2i)^2 = x_k^2 - (-4) = x_k^2 + 4\]

Thus, the product becomes:

\[P(2i) P(-2i) = 16 \left(x_1^2 + 4\right)\left(x_2^2 + 4\right)\left(x_3^2 + 4\right)\left(x_4^2 + 4\right)\]

Rearranging to isolate the desired expression:

\[\left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{P(2i)P(-2i)}{16}\]

Step 4: Compute the value of \( P(2i) \).

\[P(2i) = 4(2i)^4 + 8(2i)^3 - 17(2i)^2 - 12(2i) + 9\]

Using \( i^2 = -1, i^3 = -i, i^4 = 1 \):

\[P(2i) = 4(16 i^4) + 8(8 i^3) - 17(4 i^2) - 24i + 9\]\[P(2i) = 4(16) + 8(-8i) - 17(-4) - 24i + 9\]\[P(2i) = 64 - 64i + 68 - 24i + 9\]

Combining real and imaginary components:

\[P(2i) = (64 + 68 + 9) + (-64 - 24)i = 141 - 88i\]

Step 5: Calculate \( P(2i)P(-2i) \). Since \( P(x) \) has real coefficients, \( P(-2i) = \overline{P(2i)} = 141 + 88i \). 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 into the expression derived in 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 to the given expression to solve for \( m \).

\[\frac{27625}{16} = \frac{125}{16} m\]

Multiplying both sides by \( \frac{16}{125} \):

\[m = \frac{27625}{125}\]\[m = 221\]

The value of \( m \) is determined to be 221.