Question

fill in the blank so that the resulting statement is true.

if \\(a + bi\\) is a root of a polynomial equation with real coefficients, \\(b \
eq 0\\), then is also a root of the equation.

options:
\\(b + ai\\)
\\(a - bi\\)
\\(b - ai\\)

Explanation:

Identify the given statement and theorem

Using the Complex Numbers and Fundamental Theorem of Algebra knowledge points

$$ \text{Given root: } a + bi \quad (b eq 0) $$

$$ \text{Polynomial coefficients: } \mathbb{R} \text{ (real numbers)} $$

Apply the Complex Conjugate Root Theorem

For any polynomial equation with real coefficients, non-real complex roots must occur in conjugate pairs.

$$ \text{If } z = a + bi \text{ is a root, then its conjugate } \bar{z} = a - bi \text{ is also a root.} $$

Match with the given options

The dropdown menu displays three choices:

1. \(b + ai\)
2. \(a - bi\)
3. \(b - ai\)

The correct conjugate is \(a - bi\).

Answer:

Fill in the blank so that the resulting statement is true.

If \(a + bi\) is a root of a polynomial equation with real coefficients, \(b
eq 0\), then <blank>\(a - bi\)</blank> is also a root of the equation.