Question

If \(2,3,6\) are the roots of the polynomial \[ f(x)=x^3+ax^2+bx+c, \] where \(a,b,c\in \mathbb{C}\), then the value of \(a-c\) is

• A. \(-11\)
• B. \(36\)
• C. \(25\)
• D. \(11\)

Hint:

If the roots of a monic cubic polynomial are \(\alpha,\beta,\gamma\), then the polynomial is \[ (x-\alpha)(x-\beta)(x-\gamma). \] Expand and compare coefficients to find unknown constants.

Answer 1

The Correct Option is C

Step 1: Use Vieta's formulas.
For \(f(x)=x^3+ax^2+bx+c\) with roots 2, 3, 6: \(a = -(2+3+6) = -11\) and \(c = -(2 \cdot 3 \cdot 6) = -36\).

Step 2: Compute \(a-c\).
\[a - c = -11 - (-36) = 25.\]
\[\boxed{a - c = 25}\]