Find the equation whose are the translates of the roots of
` x^5 -4x^4 +3x^2 -4x +6=0` by ` -3`

Answer

To find the equation whose roots are the translates of the roots of the given polynomial \( x^5 - 4x^4 + 3x^2 - 4x + 6 = 0 \) by \(-3\), we can follow these steps:

### Step 1: Define the original polynomial
Let \( f(x) = x^5 - 4x^4 + 3x^2 - 4x + 6 \).

### Step 2: Translate the roots
To translate the roots of the polynomial by \(-3\), we replace \( x \) with \( x + 3 \) in the polynomial. This means we need to compute \( f(x + 3) \).

### Step 3: Calculate \( f(x + 3) \)
Now, we will substitute \( x + 3 \) into the polynomial:

\[ f(x + 3) = (x + 3)^5 - 4(x + 3)^4 + 3(x + 3)^2 - 4(x + 3) + 6 \]

### Step 4: Expand each term
Now we will expand each term:

1. **Expand \( (x + 3)^5 \)**:
\[ (x + 3)^5 = x^5 + 15x^4 + 90x^3 + 270x^2 + 243 \]

2. **Expand \( -4(x + 3)^4 \)**:
\[ -4(x + 3)^4 = -4(x^4 + 12x^3 + 54x^2 + 108x + 81) = -4x^4 - 48x^3 - 216x^2 - 432x - 324 \]

3. **Expand \( 3(x + 3)^2 \)**:
\[ 3(x + 3)^2 = 3(x^2 + 6x + 9) = 3x^2 + 18x + 27 \]

4. **Expand \( -4(x + 3) \)**:
\[ -4(x + 3) = -4x - 12 \]

5. **Constant term**:
\[ +6 \]

### Step 5: Combine all the expanded terms
Now we combine all these expanded terms:

\[ f(x + 3) = (x^5 + 15x^4 + 90x^3 + 270x^2 + 243) + (-4x^4 - 48x^3 - 216x^2 - 432x - 324) + (3x^2 + 18x + 27) + (-4x - 12) + 6 \]

### Step 6: Simplify the expression
Now we will combine like terms:

- **For \( x^5 \)**: \( 1x^5 \)
- **For \( x^4 \)**: \( 15x^4 - 4x^4 = 11x^4 \)
- **For \( x^3 \)**: \( 90x^3 - 48x^3 = 42x^3 \)
- **For \( x^2 \)**: \( 270x^2 - 216x^2 + 3x^2 = 57x^2 \)
- **For \( x \)**: \( -432x + 18x - 4x = -418x \)
- **Constant**: \( 243 - 324 + 27 - 12 + 6 = -60 \)

### Final Equation
Thus, the equation whose roots are the translates of the roots of the original polynomial by \(-3\) is:

\[ f(x + 3) = x^5 + 11x^4 + 42x^3 + 57x^2 - 418x - 60 = 0 \]