9. If the equation \( x^{3}-6 x^{2}+p x+q=0 \) has 3 equal roots, then (A) \( q=0 \) (B) \( p=0 \) (C) \( q=2 \) (D) each root \( =2 \) (E) each root \( =-2 \)

To determine the conditions under which the equation \( x^{3}-6 x^{2}+p x+q=0 \) has three equal roots, we can use the fact that if a cubic polynomial has three equal roots, it can be expressed in the form \( (x - r)^3 = 0 \), where \( r \) is the repeated root. 1. **Expand the expression**: \[ (x - r)^3 = x^3 - 3rx^2 + 3r^2x - r^3 \] This means that the coefficients of the polynomial \( x^3 - 6x^2 + px + q \) must match those of \( x^3 - 3rx^2 + 3r^2x - r^3 \). 2. **Match coefficients**: - From the coefficient of \( x^2 \): \[ -3r = -6 \implies r = 2 \] - From the coefficient of \( x \): \[ 3r^2 = p \implies p = 3(2^2) = 12 \] - From the constant term: \[ -r^3 = q \implies q = -2^3 = -8 \] 3. **Conclusion**: - The roots of the equation are \( r = 2 \), which means each root is \( 2 \). - The values of \( p \) and \( q \) are \( p = 12 \) and \( q = -8 \). Now, let's analyze the options given: - (A) \( q=0 \) → False - (B) \( p=0 \) → False - (C) \( q=2 \) → False - (D) each root \( =2 \) → True - (E) each root \( =-2 \) → False Thus, the correct answer is: **(D) each root \( =2 \)**.