Question

One of the roots of the equation, \( z^6 - 3z^4 - 16 = 0 \) is given by \( z_1 = 2 \). The value of the product of the other five roots is ...........

Hint:

To find the product of the roots, use the relationship between the coefficients of the polynomial and its roots.

Answer 1

Correct Answer: -8

Step 1: Factor the equation.
The given equation is \[ z^6 - 3z^4 - 16 = 0. \] Let \( z^2 = w \). Substituting \( w \) into the equation, we get \[ w^3 - 3w^2 - 16 = 0. \] Step 2: Solve for the roots.
Since \( z_1 = 2 \), substitute \( z = 2 \) into the original equation: \[ 2^6 - 3 \times 2^4 - 16 = 64 - 48 - 16 = 0. \] Thus, \( z_1 = 2 \) is indeed a root. Now, divide the polynomial \( w^3 - 3w^2 - 16 \) by \( w - 2 \) to find the other roots. Using synthetic or polynomial division, we get: \[ w^3 - 3w^2 - 16 = (w - 2)(w^2 - w - 8). \] Step 3: Solve for \( w \).
Now, solve \( w^2 - w - 8 = 0 \) using the quadratic formula: \[ w = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-8)}}{2(1)} = \frac{1 \pm \sqrt{1 + 32}}{2} = \frac{1 \pm \sqrt{33}}{2}. \] Thus, \( w_2 = \frac{1 + \sqrt{33}}{2} \) and \( w_3 = \frac{1 - \sqrt{33}}{2} \). Step 4: Find the product of the roots.
The product of all the roots of a polynomial is given by \( (-1)^n \times \frac{\text{constant term}}{\text{leading coefficient}} \). In this case, for the equation \( z^6 - 3z^4 - 16 = 0 \), the constant term is \( -16 \) and the leading coefficient is 1. Therefore, the product of all six roots is \[ (-1)^6 \times \frac{-16}{1} = -16. \] Since \( z_1 = 2 \), the product of the other five roots is \[ \frac{-16}{2} = -8. \] Step 5: Final Answer.
Hence, the correct answer is \( \boxed{-16} \).