Question

If $ 3 + i $ and $ 2 - \sqrt{3} $ are the roots of the equation $ f(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n $, where $ a_0, a_1, ..., a_n \in \mathbb{Z} $, then the least value of $ n $ and the value of $ a_0 $ are respectively:

• A. \( 4,\ 1 \)
• B. \( 4,\ 10 \)
• C. \( 4,\ -10 \)
• D. \( 4,\ -1 \)

Hint:

For polynomials with integer coefficients, conjugate complex and surd roots must also be included. Multiply the minimal quadratic pairs and then expand.

Answer 1

The Correct Option is B

Step 1: Use the fact that complex and irrational roots occur in conjugate pairs in polynomials with integer coefficients.
Since the polynomial has integer coefficients, the complex conjugate \( 3 - i \) must also be a root. Similarly, \( 2 + \sqrt{3} \) must also be a root.
So the roots are: \[ 3+i,\ 3-i,\ 2+\sqrt{3},\ 2-\sqrt{3}. \]
Step 2: Construct the minimal polynomial using these roots. Form the factors corresponding to the conjugate root pairs: \[ (x - (3+i))(x - (3-i)) = [(x - 3) - i][(x - 3) + i] = (x - 3)^2 + 1 = x^2 - 6x + 10, \] \[ (x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) = [(x - 2) - \sqrt{3}][(x - 2) + \sqrt{3}] = (x - 2)^2 - 3 = x^2 - 4x + 1. \]
Step 3: Multiply the two quadratic factors. \[ f(x) = (x^2 - 6x + 10)(x^2 - 4x + 1). \] Expand: First multiply: \[ x^2(x^2 - 4x + 1) = x^4 - 4x^3 + x^2, \] \[ -6x(x^2 - 4x + 1) = -6x^3 + 24x^2 - 6x, \] \[ 10(x^2 - 4x + 1) = 10x^2 - 40x + 10. \] Add all terms: \[ x^4 - 4x^3 + x^2 - 6x^3 + 24x^2 - 6x + 10x^2 - 40x + 10 = x^4 - 10x^3 + 35x^2 - 46x + 10. \] Step 4: Extract the values.
Least degree \( n = 4 \), and constant term \( a_0 = 10 \).