Find all zeroes of the polynomial `2x^4-9x^3+5x^2+3x-1` if two of its zeroes are `2+sqrt(3)` and `2-sqrt(3)`

Explanation

To find all the zeroes of the polynomial \( p(x) = 2x^4 - 9x^3 + 5x^2 + 3x - 1 \), given that two of its zeroes are \( 2 + \sqrt{3} \) and \( 2 - \sqrt{3} \), we can follow these steps:

### Step 1: Identify the known roots and form a quadratic factor
The two known roots are \( 2 + \sqrt{3} \) and \( 2 - \sqrt{3} \). We can express these roots in the factorized form:
\[ (x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) \]
This can be simplified using the difference of squares:
\[ = (x - 2 - \sqrt{3})(x - 2 + \sqrt{3}) = (x - 2)^2 - (\sqrt{3})^2 = (x - 2)^2 - 3 \]
Thus, we have:
\[ = (x - 2)^2 - 3 = x^2 - 4x + 1 \]

### Step 2: Divide the polynomial by the quadratic factor
Now we need to divide the polynomial \( p(x) \) by the quadratic factor \( d(x) = x^2 - 4x + 1 \) to find the other factor \( q(x) \).

### Step 3: Perform polynomial long division
We will divide \( p(x) \) by \( d(x) \):
1. Divide the leading term \( 2x^4 \) by \( x^2 \) to get \( 2x^2 \).
2. Multiply \( d(x) \) by \( 2x^2 \):
\[ 2x^2(x^2 - 4x + 1) = 2x^4 - 8x^3 + 2x^2 \]
3. Subtract this from \( p(x) \):
\[ (2x^4 - 9x^3 + 5x^2 + 3x - 1) - (2x^4 - 8x^3 + 2x^2) = -x^3 + 3x^2 + 3x - 1 \]
4. Repeat this process: Divide \( -x^3 \) by \( x^2 \) to get \( -x \).
5. Multiply \( d(x) \) by \( -x \):
\[ -x(x^2 - 4x + 1) = -x^3 + 4x^2 - x \]
6. Subtract:
\[ (-x^3 + 3x^2 + 3x - 1) - (-x^3 + 4x^2 - x) = -x^2 + 4x - 1 \]
7. Finally, divide \( -x^2 \) by \( x^2 \) to get \( -1 \).
8. Multiply \( d(x) \) by \( -1 \):
\[ -1(x^2 - 4x + 1) = -x^2 + 4x - 1 \]
9. Subtract:
\[ (-x^2 + 4x - 1) - (-x^2 + 4x - 1) = 0 \]

Thus, the quotient is:
\[ q(x) = 2x^2 - x - 1 \]

### Step 4: Factor the quadratic \( q(x) \)
Now we need to factor \( q(x) = 2x^2 - x - 1 \):
1. We can factor it as:
\[ (2x + 1)(x - 1) = 0 \]

### Step 5: Find the remaining roots
Setting each factor to zero gives us the remaining roots:
1. \( 2x + 1 = 0 \) leads to \( x = -\frac{1}{2} \)
2. \( x - 1 = 0 \) leads to \( x = 1 \)

### Final Result
The complete set of zeroes of the polynomial \( p(x) \) is:
\[ 2 + \sqrt{3}, \quad 2 - \sqrt{3}, \quad -\frac{1}{2}, \quad 1 \]