Obtian all other zeroes of `(x^(4)+4x^(3)-2x^(2)-20x-15)` if two of its zeroes are ` :' sqrt(5)` and `-sqrt(5)` .

Explanation

To find all the zeros of the polynomial \( P(x) = x^4 + 4x^3 - 2x^2 - 20x - 15 \) given that two of its zeros are \( \sqrt{5} \) and \( -\sqrt{5} \), we can follow these steps:

### Step 1: Identify the factors
Since \( \sqrt{5} \) and \( -\sqrt{5} \) are zeros, we can express the polynomial as having factors \( (x - \sqrt{5}) \) and \( (x + \sqrt{5}) \). The product of these factors is:
\[ (x - \sqrt{5})(x + \sqrt{5}) = x^2 - 5 \]
Thus, \( x^2 - 5 \) is a factor of \( P(x) \).

### Step 2: Divide the polynomial by the factor
Now, we will divide \( P(x) \) by \( x^2 - 5 \) to find the other factor. We can perform polynomial long division.

1. Divide the leading term of \( P(x) \) by the leading term of \( x^2 - 5 \):
\[ \frac{x^4}{x^2} = x^2 \]
2. Multiply \( x^2 \) by \( x^2 - 5 \):
\[ x^2(x^2 - 5) = x^4 - 5x^2 \]
3. Subtract this from \( P(x) \):
\[ P(x) - (x^4 - 5x^2) = (4x^3 + 3x^2 - 20x - 15) \]

4. Now, divide the leading term \( 4x^3 \) by \( x^2 \):
\[ \frac{4x^3}{x^2} = 4x \]
5. Multiply \( 4x \) by \( x^2 - 5 \):
\[ 4x(x^2 - 5) = 4x^3 - 20x \]
6. Subtract:
\[ (4x^3 + 3x^2 - 20x - 15) - (4x^3 - 20x) = 3x^2 - 15 \]

7. Divide \( 3x^2 \) by \( x^2 \):
\[ \frac{3x^2}{x^2} = 3 \]
8. Multiply \( 3 \) by \( x^2 - 5 \):
\[ 3(x^2 - 5) = 3x^2 - 15 \]
9. Subtract:
\[ (3x^2 - 15) - (3x^2 - 15) = 0 \]

### Step 3: Write the polynomial in factored form
After the division, we find:
\[ P(x) = (x^2 - 5)(x^2 + 4x + 3) \]

### Step 4: Factor the remaining quadratic
Now we need to factor \( x^2 + 4x + 3 \):
1. We can factor it as:
\[ x^2 + 4x + 3 = (x + 1)(x + 3) \]

### Step 5: Find all zeros
Thus, we can write:
\[ P(x) = (x - \sqrt{5})(x + \sqrt{5})(x + 1)(x + 3) \]
The zeros of \( P(x) \) are:
1. \( x = \sqrt{5} \)
2. \( x = -\sqrt{5} \)
3. \( x = -1 \)
4. \( x = -3 \)

### Conclusion
The other zeros of the polynomial \( P(x) \) are \( -1 \) and \( -3 \).

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