Solve the equation given that it has multiple root `x^4 +2x^3 -3x^2 -4x +4=0`

Answer

To solve the equation \( x^4 + 2x^3 - 3x^2 - 4x + 4 = 0 \) given that it has multiple roots, we will follow these steps:

### Step 1: Identify a root
Since the equation has multiple roots, we can start by checking for rational roots using the Rational Root Theorem. We can test \( x = 1 \):
\[ f(1) = 1^4 + 2(1^3) - 3(1^2) - 4(1) + 4 = 1 + 2 - 3 - 4 + 4 = 0 \]
Thus, \( x = 1 \) is a root.

### Step 2: Factor out \( (x - 1)^2 \)
Since \( x = 1 \) is a multiple root, we can factor the polynomial as \( (x - 1)^2 \) times another polynomial. We will perform polynomial long division of \( x^4 + 2x^3 - 3x^2 - 4x + 4 \) by \( (x - 1)^2 \).

First, we find \( (x - 1)^2 = x^2 - 2x + 1 \).

Now, we divide \( x^4 + 2x^3 - 3x^2 - 4x + 4 \) by \( x^2 - 2x + 1 \).

1. Divide the leading term: \( x^4 \div x^2 = x^2 \).
2. Multiply: \( x^2(x^2 - 2x + 1) = x^4 - 2x^3 + x^2 \).
3. Subtract:
\[ (x^4 + 2x^3 - 3x^2 - 4x + 4) - (x^4 - 2x^3 + x^2) = 4x^3 - 4x^2 - 4x + 4 \]

4. Divide the leading term: \( 4x^3 \div x^2 = 4x \).
5. Multiply: \( 4x(x^2 - 2x + 1) = 4x^3 - 8x^2 + 4x \).
6. Subtract:
\[ (4x^3 - 4x^2 - 4x + 4) - (4x^3 - 8x^2 + 4x) = 4x^2 + 4 \]

7. Divide the leading term: \( 4x^2 \div x^2 = 4 \).
8. Multiply: \( 4(x^2 - 2x + 1) = 4x^2 - 8x + 4 \).
9. Subtract:
\[ (4x^2 + 4) - (4x^2 - 8x + 4) = 8x \]

So, we have:
\[ x^4 + 2x^3 - 3x^2 - 4x + 4 = (x - 1)^2 (x^2 + 4x + 4) \]

### Step 3: Factor the quadratic
Now, we can factor \( x^2 + 4x + 4 \):
\[ x^2 + 4x + 4 = (x + 2)^2 \]

### Step 4: Write the complete factorization
Thus, the complete factorization of the polynomial is:
\[ (x - 1)^2 (x + 2)^2 = 0 \]

### Step 5: Solve for the roots
Setting each factor to zero gives us the roots:
1. \( (x - 1)^2 = 0 \) implies \( x = 1 \) (multiplicity 2).
2. \( (x + 2)^2 = 0 \) implies \( x = -2 \) (multiplicity 2).

### Final Roots
The roots of the equation \( x^4 + 2x^3 - 3x^2 - 4x + 4 = 0 \) are:
- \( x = 1 \) (multiplicity 2)
- \( x = -2 \) (multiplicity 2)