Question

If the equation \(2x^3 + 5x^2 - 4x - 12 = 0\) has a repeated root, then the constant term of the quadratic equation whose roots are the distinct roots of the given equation is:

• A. \( \mathbf{-6} \)
• B. \( -5 \)
• C. \( -4 \)
• D. \( -2 \)

Hint:

If a root is repeated, factor it out, then form a new equation from the distinct roots.

Answer 1

The Correct Option is A

Let the repeated root be \( \alpha \), and distinct roots be \( \beta \), \( \gamma \). Then the cubic factors as: \[ (x - \alpha)^2(x - \beta) = 2x^3 + 5x^2 - 4x - 12 \] Divide the polynomial by \( (x - \alpha)^2 \) using long division (or synthetic division), or assume form: \[ 2x^3 + 5x^2 - 4x - 12 = (x - \alpha)^2(x - \beta) \Rightarrow \text{Quadratic: } (x - \beta)(x - \gamma) = x^2 - (\beta + \gamma)x + \beta\gamma \] From this, constant term = product of roots = \( \beta\gamma \) Once polynomial is factorized or the distinct quadratic roots found, their product (constant term) = \( \boxed{-6} \)