Find the equation whose are the translates of the roots of
`3x^5 -5x^3 +7 =0` by 4

Answer

To find the equation whose roots are the translates of the roots of the given polynomial \(3x^5 - 5x^3 + 7 = 0\) by 4, we can follow these steps:

### Step 1: Define the Given Polynomial
Let \(f(x) = 3x^5 - 5x^3 + 7\).

### Step 2: Translate the Roots
Since we want the roots translated by 4, we set \(x = y + 4\). This means we will replace \(x\) in \(f(x)\) with \(y + 4\).

### Step 3: Substitute into the Polynomial
Now, we substitute \(x\) in the polynomial:
\[ f(y + 4) = 3(y + 4)^5 - 5(y + 4)^3 + 7 \]

### Step 4: Expand the Polynomial
We need to expand \(f(y + 4)\):
1. **Expand \((y + 4)^5\)**:
\[ (y + 4)^5 = y^5 + 5 \cdot 4y^4 + 10 \cdot 16y^3 + 10 \cdot 64y^2 + 5 \cdot 256y + 1024 \]
\[ = y^5 + 20y^4 + 160y^3 + 640y^2 + 1280y + 1024 \]

2. **Expand \((y + 4)^3\)**:
\[ (y + 4)^3 = y^3 + 3 \cdot 4y^2 + 3 \cdot 16y + 64 \]
\[ = y^3 + 12y^2 + 48y + 64 \]

### Step 5: Substitute Back into the Polynomial
Now substituting back into \(f(y + 4)\):
\[ f(y + 4) = 3(y^5 + 20y^4 + 160y^3 + 640y^2 + 1280y + 1024) - 5(y^3 + 12y^2 + 48y + 64) + 7 \]

### Step 6: Combine Like Terms
Now we combine the terms:
\[ = 3y^5 + 60y^4 + 480y^3 + 1920y^2 + 3840y + 3072 - 5y^3 - 60y^2 - 240y - 320 + 7 \]
\[ = 3y^5 + 60y^4 + (480 - 5)y^3 + (1920 - 60)y^2 + (3840 - 240)y + (3072 - 320 + 7) \]
\[ = 3y^5 + 60y^4 + 475y^3 + 1860y^2 + 3600y + 2759 \]

### Step 7: Write the Final Equation
Thus, the required equation whose roots are the translates of the roots of the original polynomial by 4 is:
\[ 3y^5 + 60y^4 + 475y^3 + 1860y^2 + 3600y + 2759 = 0 \]

### Summary
The final equation is:
\[ 3x^5 + 60x^4 + 475x^3 + 1860x^2 + 3600x + 2759 = 0 \]