Find the polynomial with rational coefficients and whose roots are
`1 +- sqrt(3),2,5`

Answer

To find the polynomial with rational coefficients whose roots are \(1 + \sqrt{3}\), \(1 - \sqrt{3}\), \(2\), and \(5\), we can follow these steps:

### Step 1: Identify the roots
The roots given are:
- \( \alpha = 1 + \sqrt{3} \)
- \( \beta = 1 - \sqrt{3} \)
- \( \gamma = 2 \)
- \( \delta = 5 \)

### Step 2: Write the polynomial in factored form
The polynomial can be expressed as:
\[ P(x) = (x - \alpha)(x - \beta)(x - \gamma)(x - \delta) \]
Substituting the roots, we have:
\[ P(x) = (x - (1 + \sqrt{3}))(x - (1 - \sqrt{3}))(x - 2)(x - 5) \]

### Step 3: Simplify the first two factors
First, we simplify the product of the first two factors:
\[ (x - (1 + \sqrt{3}))(x - (1 - \sqrt{3})) = (x - 1 - \sqrt{3})(x - 1 + \sqrt{3}) \]
This is a difference of squares:
\[ = (x - 1)^2 - (\sqrt{3})^2 = (x - 1)^2 - 3 \]
Expanding this gives:
\[ = x^2 - 2x + 1 - 3 = x^2 - 2x - 2 \]

### Step 4: Simplify the last two factors
Now, we simplify the product of the last two factors:
\[ (x - 2)(x - 5) = x^2 - 7x + 10 \]

### Step 5: Combine the two results
Now we combine the two quadratic factors:
\[ P(x) = (x^2 - 2x - 2)(x^2 - 7x + 10) \]

### Step 6: Expand the polynomial
Now we expand the polynomial:
\[ P(x) = x^2(x^2 - 7x + 10) - 2x(x^2 - 7x + 10) - 2(x^2 - 7x + 10) \]
Calculating each term:
1. \(x^2(x^2 - 7x + 10) = x^4 - 7x^3 + 10x^2\)
2. \(-2x(x^2 - 7x + 10) = -2x^3 + 14x^2 - 20x\)
3. \(-2(x^2 - 7x + 10) = -2x^2 + 14x - 20\)

Combining all these gives:
\[ P(x) = x^4 - 7x^3 + 10x^2 - 2x^3 + 14x^2 - 20x - 2x^2 + 14x - 20 \]
\[ = x^4 - 9x^3 + (10x^2 + 14x^2 - 2x^2) + (-20x + 14x) - 20 \]
\[ = x^4 - 9x^3 + 22x^2 - 6x - 20 \]

### Final Result
Thus, the polynomial with rational coefficients whose roots are \(1 + \sqrt{3}\), \(1 - \sqrt{3}\), \(2\), and \(5\) is:
\[ \boxed{x^4 - 9x^3 + 22x^2 - 6x - 20} \]