Find the multiple roots of
`x^4 - 6x^3 + 13 x^2 - 24 x +36 =0`

Answer

To find the multiple roots of the polynomial \( f(x) = x^4 - 6x^3 + 13x^2 - 24x + 36 \), we will follow these steps:

### Step 1: Identify the polynomial
We start with the polynomial:
\[ f(x) = x^4 - 6x^3 + 13x^2 - 24x + 36 \]

### Step 2: Use the Rational Root Theorem
We will look for rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (36), which are:
\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 36 \]

### Step 3: Test possible roots
We will test \( x = 3 \):
\[ f(3) = 3^4 - 6(3^3) + 13(3^2) - 24(3) + 36 \]
Calculating each term:
\[ = 81 - 162 + 117 - 72 + 36 \]
\[ = 81 - 162 + 117 - 72 + 36 = 0 \]
Since \( f(3) = 0 \), \( x - 3 \) is a factor of \( f(x) \).

### Step 4: Polynomial long division
Now we will divide \( f(x) \) by \( x - 3 \):
\[ \text{Performing long division:} \]
1. Divide \( x^4 \) by \( x \) to get \( x^3 \).
2. Multiply \( x^3 \) by \( x - 3 \) to get \( x^4 - 3x^3 \).
3. Subtract:
\[ (x^4 - 6x^3) - (x^4 - 3x^3) = -3x^3 \]
4. Bring down the next term \( + 13x^2 \):
\[ -3x^3 + 13x^2 = -3x^3 + 13x^2 \]
5. Divide \( -3x^3 \) by \( x \) to get \( -3x^2 \).
6. Multiply \( -3x^2 \) by \( x - 3 \) to get \( -3x^3 + 9x^2 \).
7. Subtract:
\[ (-3x^3 + 13x^2) - (-3x^3 + 9x^2) = 4x^2 \]
8. Bring down \( -24x \):
\[ 4x^2 - 24x \]
9. Divide \( 4x^2 \) by \( x \) to get \( 4x \).
10. Multiply \( 4x \) by \( x - 3 \) to get \( 4x^2 - 12x \).
11. Subtract:
\[ (4x^2 - 24x) - (4x^2 - 12x) = -12x \]
12. Bring down \( +36 \):
\[ -12x + 36 \]
13. Divide \( -12x \) by \( x \) to get \( -12 \).
14. Multiply \( -12 \) by \( x - 3 \) to get \( -12x + 36 \).
15. Subtract:
\[ (-12x + 36) - (-12x + 36) = 0 \]

Thus, we have:
\[ f(x) = (x - 3)(x^3 - 3x^2 + 4x - 12) \]

### Step 5: Factor the cubic polynomial
Next, we will factor \( x^3 - 3x^2 + 4x - 12 \). We will test \( x = 3 \) again:
\[ f(3) = 3^3 - 3(3^2) + 4(3) - 12 \]
Calculating:
\[ = 27 - 27 + 12 - 12 = 0 \]
So, \( x - 3 \) is also a factor of \( x^3 - 3x^2 + 4x - 12 \).

### Step 6: Divide again
Now we divide \( x^3 - 3x^2 + 4x - 12 \) by \( x - 3 \):
1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x - 3 \) to get \( x^3 - 3x^2 \).
3. Subtract:
\[ (x^3 - 3x^2) - (x^3 - 3x^2) = 0 \]
4. Bring down \( +4x - 12 \):
5. Divide \( 4x \) by \( x \) to get \( 4 \).
6. Multiply \( 4 \) by \( x - 3 \) to get \( 4x - 12 \).
7. Subtract:
\[ (4x - 12) - (4x - 12) = 0 \]

Thus, we have:
\[ f(x) = (x - 3)^2 (x^2 + 4) \]

### Step 7: Solve for roots of \( x^2 + 4 = 0 \)
Setting \( x^2 + 4 = 0 \):
\[ x^2 = -4 \implies x = \pm 2i \]

### Conclusion
The multiple roots of the equation \( x^4 - 6x^3 + 13x^2 - 24x + 36 = 0 \) are:
\[ x = 3 \text{ (with multiplicity 2)}, \quad x = 2i, \quad x = -2i \]