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 $$ =-2 $$

Question

UpStudy AI · Accepted Answer

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 $$ =2 $$**.
Each root is equal to 2.

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 \( =-2 \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions