Finding the Value of \(|z|\) for the Given Complex Number Equation

We are given the complex number equation \(iz^3 + z^2 - z + i = 0\). Our goal is to find the value of \(|z|\) that satisfies this equation. This involves analyzing the roots of the complex number equation.

Solving the Cubic Equation

To find the possible values of \(z\) and subsequently the value of \(|z|\), we need to solve this cubic equation. Solving cubic equations, especially those with complex coefficients, often involves finding one root and then factoring the polynomial.

Factoring the Polynomial \(iz^3 + z^2 - z + i\)

Let's try to find a simple root by inspection. If we test \(z=i\), we substitute it into the equation:

\(i(i)^3 + (i)^2 - i + i = i(-i) + (-1) - i + i = -i^2 - 1 + 0 = 1 - 1 = 0\)

Since substituting \(z=i\) satisfies the equation, \(z=i\) is a root. This means that \((z-i)\) is a factor of the polynomial \(iz^3 + z^2 - z + i\). We can use polynomial division or factoring by grouping to find the other factor.

Let's factor the polynomial: \(iz^3 + z^2 - z + i = iz^3 - i + z^2 - z + 2i\). This doesn't seem straightforward.

Knowing that \((z-i)\) is a factor, we can perform polynomial division:

\((iz^3 + z^2 - z + i) \div (z-i) = iz^2 + 1\)

So, the original equation can be factored as:

\((z-i)(iz^2 + 1) = 0\)

We can verify this factoring polynomial step by expanding the product: \((z-i)(iz^2 + 1) = z(iz^2+1) - i(iz^2+1) = iz^3 + z - i^2 z^2 - i = iz^3 + z + z^2 - i = iz^3 + z^2 + z - i\). This is not the original polynomial.

Let's re-do the polynomial division carefully:

\((iz^3 + z^2 - z + i) \div (z-i)\)

• Divide \(iz^3\) by \(z\): \(iz^2\). Multiply \((z-i)(iz^2) = iz^3 - i^2 z^2 = iz^3 + z^2\). Subtract from the original polynomial: \((iz^3 + z^2 - z + i) - (iz^3 + z^2) = -z + i\).
• Divide \(-z\) by \(z\): \(-1\). Multiply \((z-i)(-1) = -z + i\). Subtract from the remainder: \((-z + i) - (-z + i) = 0\).

The quotient is \(iz^2 - 1\). The correct factorization is indeed \((z-i)(iz^2 - 1) = 0\). Let's verify again:

\((z-i)(iz^2 - 1) = z(iz^2) + z(-1) - i(iz^2) - i(-1) = iz^3 - z - i^2 z^2 + i = iz^3 - z + z^2 + i = iz^3 + z^2 - z + i\). This matches the original complex number equation.

Determining the Value of \(|z|\) from the Roots

The roots of the equation are found by setting each factor to zero.

Case 1: The Root from \(z - i = 0\)

Setting the first factor to zero gives \(z - i = 0\), which means \(z = i\).

The modulus of this root is \(|z| = |i|\).

\(|i| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1\)

So, one possible value of \(|z|\) is 1.

Case 2: The Roots from the Equation \(iz^2 - 1 = 0\)

Setting the second factor to zero gives a quadratic equation in \(z\): \(iz^2 - 1 = 0\).

Solving for \(z^2\): \(iz^2 = 1 \implies z^2 = \frac{1}{i}\).

Since \(\frac{1}{i} = \frac{1}{i} \times \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{1} = -i\), we have \(z^2 = -i\).

To find \(z\), we need to find the square roots of \(-i\). We can express \(-i\) in polar form. The modulus of \(-i\) is \(|-i| = \sqrt{0^2 + (-1)^2} = 1\). The argument of \(-i\) is \(-\frac{\pi}{2}\) (or \(\frac{3\pi}{2}\)).

So, \(-i = 1 \cdot e^{i(-\pi/2 + 2k\pi)}\) for integer \(k\).

If \(z = re^{i\theta}\), then \(z^2 = r^2 e^{i2\theta}\). Setting \(z^2 = -i\):

\(r^2 e^{i2\theta} = 1 \cdot e^{i(-\pi/2 + 2k\pi)}\)

Comparing moduli, \(r^2 = 1 \implies r = 1\) (since modulus must be non-negative).

Comparing arguments, \(2\theta = -\frac{\pi}{2} + 2k\pi \implies \theta = -\frac{\pi}{4} + k\pi\).

For \(k=0\), \(\theta_1 = -\frac{\pi}{4}\). The root is \(z_1 = 1 \cdot e^{-i\pi/4} = \cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}) = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}\).

The modulus of \(z_1\) is \(|z_1| = \sqrt{(\frac{1}{\sqrt{2}})^2 + (-\frac{1}{\sqrt{2}})^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1\).

For \(k=1\), \(\theta_2 = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}\). The root is \(z_2 = 1 \cdot e^{i3\pi/4} = \cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}) = -\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\).

The modulus of \(z_2\) is \(|z_2| = \sqrt{(-\frac{1}{\sqrt{2}})^2 + (\frac{1}{\sqrt{2}})^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1\).

Conclusion: The Value of \(|z|\)

The complex number equation \(iz^3 + z^2 - z + i = 0\) has three roots: \(z=i\), \(z=\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}\), and \(z=-\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\). We calculated the modulus for each of these roots:

• \(|i| = 1\)
• \(|\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}| = 1\)
• \(|-\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}| = 1\)

All the roots have a modulus equal to 1. Therefore, the value of \(|z|\) is 1. This value is present among the given options.

Note that the modulus of a complex number cannot be negative, ruling out option -1. Our calculation shows that for any root of this complex number equation, the value of \(|z|\) is 1.