Find the equation whose are the translates of the roots of
`x^5 +4x^3 -x^2 +11=0` by -3

Answer

To find the equation whose roots are the translates of the roots of the polynomial \( x^5 + 4x^3 - x^2 + 11 = 0 \) by -3, we will follow these steps:

### Step 1: Define the Original Polynomial
Let \( f(x) = x^5 + 4x^3 - x^2 + 11 \).

### Step 2: Translate the Roots
If the roots of \( f(x) = 0 \) are \( r_1, r_2, r_3, r_4, r_5 \), then the new roots after translating by -3 will be \( r_1 - 3, r_2 - 3, r_3 - 3, r_4 - 3, r_5 - 3 \).

### Step 3: Substitute \( x + 3 \) into the Polynomial
To find the new polynomial with roots \( r_i - 3 \), we substitute \( x + 3 \) into the original polynomial:
\[ f(x + 3) = (x + 3)^5 + 4(x + 3)^3 - (x + 3)^2 + 11 \]

### Step 4: Expand \( f(x + 3) \)
Now we will expand each term:
1. **Expand \( (x + 3)^5 \)**:
\[ (x + 3)^5 = x^5 + 15x^4 + 135x^3 + 405x^2 + 243x + 243 \]

2. **Expand \( 4(x + 3)^3 \)**:
\[ 4(x + 3)^3 = 4(x^3 + 9x^2 + 27x + 27) = 4x^3 + 36x^2 + 108x + 108 \]

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

4. **Combine all the terms**:
Now we combine all the expanded terms:
\[ f(x + 3) = (x^5 + 15x^4 + 135x^3 + 405x^2 + 243x + 243) + (4x^3 + 36x^2 + 108x + 108) - (x^2 + 6x + 9) + 11 \]

### Step 5: Collect Like Terms
Now, we will collect like terms:
- \( x^5 \) term: \( 1x^5 \)
- \( x^4 \) term: \( 15x^4 \)
- \( x^3 \) term: \( 135x^3 + 4x^3 = 139x^3 \)
- \( x^2 \) term: \( 405x^2 + 36x^2 - x^2 = 440x^2 \)
- \( x \) term: \( 243x + 108x - 6x = 345x \)
- Constant term: \( 243 + 108 - 9 + 11 = 353 \)

Thus, we have:
\[ f(x + 3) = x^5 + 15x^4 + 139x^3 + 440x^2 + 345x + 353 \]

### Step 6: Set the New Polynomial to Zero
The required polynomial whose roots are the translates of the roots of the original polynomial by -3 is:
\[ x^5 + 15x^4 + 139x^3 + 440x^2 + 345x + 353 = 0 \]

### Final Answer
The equation is:
\[ x^5 + 15x^4 + 139x^3 + 440x^2 + 345x + 353 = 0 \]

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