Find the algebraic equation of degree 4 whose roots are the translates of the roots
`4x^4+32x^3+83x^2+76x+21=0 ` by 2.

Answer

To find the algebraic equation of degree 4 whose roots are translates of the roots of the polynomial \(4x^4 + 32x^3 + 83x^2 + 76x + 21 = 0\) by 2, we will follow these steps:

### Step 1: Identify the original polynomial
The original polynomial is:
\[ f(x) = 4x^4 + 32x^3 + 83x^2 + 76x + 21 \]

### Step 2: Translate the roots
To find the new polynomial whose roots are the translates of the original roots by 2, we need to evaluate \(f(x - 2)\).

### Step 3: Substitute \(x - 2\) into the polynomial
We will substitute \(x - 2\) into the polynomial \(f(x)\):
\[ f(x - 2) = 4(x - 2)^4 + 32(x - 2)^3 + 83(x - 2)^2 + 76(x - 2) + 21 \]

### Step 4: Expand each term
Now we will expand each term one by one.

1. **Expand \( (x - 2)^4 \)**:
\[ (x - 2)^4 = x^4 - 8x^3 + 24x^2 - 32x + 16 \]
Therefore,
\[ 4(x - 2)^4 = 4(x^4 - 8x^3 + 24x^2 - 32x + 16) = 4x^4 - 32x^3 + 96x^2 - 128x + 64 \]

2. **Expand \( (x - 2)^3 \)**:
\[ (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \]
Therefore,
\[ 32(x - 2)^3 = 32(x^3 - 6x^2 + 12x - 8) = 32x^3 - 192x^2 + 384x - 256 \]

3. **Expand \( (x - 2)^2 \)**:
\[ (x - 2)^2 = x^2 - 4x + 4 \]
Therefore,
\[ 83(x - 2)^2 = 83(x^2 - 4x + 4) = 83x^2 - 332x + 332 \]

4. **Expand \( (x - 2) \)**:
\[ 76(x - 2) = 76x - 152 \]

### Step 5: Combine all expanded terms
Now, we will combine all the expanded terms:
\[ f(x - 2) = (4x^4 - 32x^3 + 96x^2 - 128x + 64) + (32x^3 - 192x^2 + 384x - 256) + (83x^2 - 332x + 332) + (76x - 152) + 21 \]

Combining like terms:
- \(x^4\) term: \(4x^4\)
- \(x^3\) term: \(-32x^3 + 32x^3 = 0\)
- \(x^2\) term: \(96x^2 - 192x^2 + 83x^2 = -13x^2\)
- \(x\) term: \(-128x + 384x - 332x + 76x = 0\)
- Constant term: \(64 - 256 + 332 - 152 + 21 = 9\)

### Step 6: Write the new polynomial
Thus, we have:
\[ f(x - 2) = 4x^4 - 13x^2 + 9 \]

### Final Result
The algebraic equation of degree 4 whose roots are the translates of the roots of the original polynomial by 2 is:
\[ 4x^4 - 13x^2 + 9 = 0 \]

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