Find the polynomials `u(x) ` and `v(x)` such that `(x^(4) -1) * u(x) + (x^(7) -1) * v(x) = (x-1)`.

Answer

To find the polynomials \( u(x) \) and \( v(x) \) such that

\[ (x^4 - 1) u(x) + (x^7 - 1) v(x) = (x - 1), \]

we can follow these steps:

### Step 1: Factor the expressions
First, we can factor \( x^4 - 1 \) and \( x^7 - 1 \).

\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) \]

\[ x^7 - 1 = (x - 1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) \]

### Step 2: Substitute into the equation
Substituting the factored forms into the original equation gives:

\[ ((x - 1)(x^2 - 1)(x^2 + 1)) u(x) + ((x - 1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)) v(x) = (x - 1) \]

### Step 3: Factor out \( (x - 1) \)
We can factor out \( (x - 1) \) from both sides:

\[ (x - 1) \left( (x^2 - 1)(x^2 + 1) u(x) + (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) v(x) \right) = (x - 1) \]

### Step 4: Divide both sides by \( (x - 1) \)
Assuming \( x \neq 1 \), we can divide both sides by \( (x - 1) \):

\[ (x^2 - 1)(x^2 + 1) u(x) + (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) v(x) = 1 \]

### Step 5: Choose suitable polynomials
To satisfy this equation, we can choose:

1. Let \( v(x) = -x \)
2. Then we can find \( u(x) \).

Substituting \( v(x) = -x \) into the equation gives:

\[ (x^2 - 1)(x^2 + 1) u(x) - x(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) = 1 \]

### Step 6: Solve for \( u(x) \)
Now we need to simplify and solve for \( u(x) \).

After some calculations, we find that:

\[ u(x) = x^4 + 1 \]

### Final Result
Thus, the polynomials are:

\[ u(x) = x^4 + 1, \quad v(x) = -x \]