Question

If \( P(x) = 0 \) is a polynomial of least degree with integer coefficients and \( \sqrt{2} + \sqrt{3}i \) is one of its roots, then that equation is:

• A. \( x^6 - 2x^4 + 2x^2 - 25 = 0 \)
• B. \( x^5 + 3x^4 + 2x^2 + 24 = 0 \)
• C. \( x^4 + 2x^2 + 25 = 0 \)
• D. \( x^4 - 2x^2 + 25 = 0 \)

Hint:

When a root involves both a square root and imaginary part, include all conjugate roots to ensure real polynomial with integer coefficients.

Answer 1

The Correct Option is C

Step 1: Roots required for minimal integer polynomial
If \( \sqrt{2} + \sqrt{3}i \) is a root, then all of the following must be roots: \[ \sqrt{2} + \sqrt{3}i, \quad \sqrt{2} - \sqrt{3}i, \quad -\sqrt{2} + \sqrt{3}i, \quad -\sqrt{2} - \sqrt{3}i \]
Step 2: Form quadratic with each conjugate pair
First pair: Let \( x = \sqrt{2} + \sqrt{3}i \Rightarrow x - \sqrt{2} = \sqrt{3}i \Rightarrow (x - \sqrt{2})^2 = -3 \Rightarrow x^2 - 2\sqrt{2}x + 2 = -3 \) → leads to a quadratic. Eventually multiplying conjugate quadratics leads to: \[ x^4 + 2x^2 + 25 = 0 \]