If the equation ` x^(5) - 10 a^(3)x^(2) + b^(4)x + c ^(5) = 0` has three equal roots, then

Options

A. `2b^(2) - 10 a^(3)x^(2)+c^(5)=0`
B. `6a^(5) +c^(5)=0`
C. `2c^(2) - 10 a^(3)b^(2)+b^(4)c^(5)=0`
D. `b^(4)= 15a^(4)`

Answer

To solve the problem, we need to analyze the given polynomial equation and apply the conditions for it to have three equal roots. Let's denote the polynomial as:

\[ f(x) = x^5 - 10a^3 x^2 + b^4 x + c^5 \]

### Step 1: Identify the roots
Since the equation has three equal roots, we can denote these roots as \( \alpha \). Therefore, we can express the polynomial as:

\[ f(x) = (x - \alpha)^3 (x - \delta)(x - \phi) \]

where \( \delta \) and \( \phi \) are the other two roots.

### Step 2: Differentiate the function
To find the conditions for the roots, we need to differentiate the function:

\[ f'(x) = 5x^4 - 20a^3 x + b^4 \]

### Step 3: Apply the conditions for equal roots
For \( f(x) \) to have a root of multiplicity 3 at \( x = \alpha \), we require:

1. \( f(\alpha) = 0 \)
2. \( f'(\alpha) = 0 \)
3. \( f''(\alpha) = 0 \)

### Step 4: Calculate the second derivative
Now, we differentiate \( f'(x) \) to find \( f''(x) \):

\[ f''(x) = 20x^3 - 20a^3 \]

### Step 5: Set up equations
Now we can set up the equations based on the conditions:

1. From \( f'(\alpha) = 0 \):
\[ 5\alpha^4 - 20a^3 \alpha + b^4 = 0 \]

2. From \( f''(\alpha) = 0 \):
\[ 20\alpha^3 - 20a^3 = 0 \]
This simplifies to:
\[ \alpha^3 = a^3 \]
Hence, we have:
\[ \alpha = a \]

### Step 6: Substitute \( \alpha \) into the first derivative equation
Substituting \( \alpha = a \) into the first derivative equation:

\[ 5a^4 - 20a^3 a + b^4 = 0 \]
This simplifies to:
\[ 5a^4 - 20a^4 + b^4 = 0 \]
Thus:
\[ b^4 = 15a^4 \]

### Step 7: Substitute \( \alpha \) into the original function
Now, substitute \( \alpha = a \) into the original function \( f(x) \):

\[ f(a) = a^5 - 10a^3 a^2 + b^4 a + c^5 = 0 \]
This simplifies to:
\[ a^5 - 10a^5 + 15a^4 a + c^5 = 0 \]
Thus:
\[ a^5 - 10a^5 + 15a^5 + c^5 = 0 \]
This gives us:
\[ 6a^5 + c^5 = 0 \]
So, we have:
\[ c^5 = -6a^5 \]

### Final Results
Thus, the conditions for the polynomial to have three equal roots are:

1. \( b^4 = 15a^4 \)
2. \( c^5 = -6a^5 \)