If 1 and -1 are zeroes of the polynomial `x^(3)+5x^(2)-x-5`, then find all zeroes of the given polynomial.

Explanation

To find all the zeroes of the polynomial \( p(x) = x^3 + 5x^2 - x - 5 \) given that \( 1 \) and \( -1 \) are zeroes, we can follow these steps:

### Step 1: Identify the given polynomial
We start with the polynomial:
\[ p(x) = x^3 + 5x^2 - x - 5 \]

### Step 2: Use the given zeroes to find factors
Since \( 1 \) and \( -1 \) are zeroes of the polynomial, we can say that \( (x - 1) \) and \( (x + 1) \) are factors of \( p(x) \).

### Step 3: Form a quadratic factor
The product of the two factors gives us:
\[ (x - 1)(x + 1) = x^2 - 1 \]
Thus, we can express \( p(x) \) as:
\[ p(x) = (x^2 - 1) \cdot q(x) \]
where \( q(x) \) is another polynomial that we need to determine.

### Step 4: Perform polynomial long division
Now we will divide \( p(x) \) by \( x^2 - 1 \):
1. Divide the leading term \( x^3 \) by \( x^2 \) to get \( x \).
2. Multiply \( x \) by \( x^2 - 1 \) to get \( x^3 - x \).
3. Subtract \( (x^3 - x) \) from \( p(x) \):
\[ (x^3 + 5x^2 - x - 5) - (x^3 - x) = 5x^2 - 5 \]
4. Now divide \( 5x^2 \) by \( x^2 \) to get \( 5 \).
5. Multiply \( 5 \) by \( x^2 - 1 \) to get \( 5x^2 - 5 \).
6. Subtract \( (5x^2 - 5) \) from \( 5x^2 - 5 \):
\[ (5x^2 - 5) - (5x^2 - 5) = 0 \]

Thus, we have:
\[ p(x) = (x^2 - 1)(x + 5) \]

### Step 5: Factor the quadratic
Now we can rewrite \( x^2 - 1 \) as:
\[ x^2 - 1 = (x - 1)(x + 1) \]
So we can express \( p(x) \) as:
\[ p(x) = (x - 1)(x + 1)(x + 5) \]

### Step 6: Find all zeroes
Now we set each factor to zero:
1. \( x - 1 = 0 \) gives \( x = 1 \)
2. \( x + 1 = 0 \) gives \( x = -1 \)
3. \( x + 5 = 0 \) gives \( x = -5 \)

### Conclusion
The zeroes of the polynomial \( p(x) = x^3 + 5x^2 - x - 5 \) are:
\[ x = 1, \quad x = -1, \quad x = -5 \]