Form the polynomial with rational coefficients whose roots are
` 1+ 5i ,5-i`

Answer

To form a polynomial with rational coefficients whose roots are \(1 + 5i\) and \(5 - i\), we need to follow these steps:

### Step 1: Identify the roots
Let the roots be:
\[ \alpha = 1 + 5i, \quad \beta = 5 - i \]

### Step 2: Write the polynomial in factored form
The polynomial can be expressed using its roots as:
\[ P(x) = (x - \alpha)(x - \beta) \]
Substituting the values of \(\alpha\) and \(\beta\):
\[ P(x) = (x - (1 + 5i))(x - (5 - i)) \]

### Step 3: Simplify the expression
Expanding the polynomial:
\[ P(x) = (x - 1 - 5i)(x - 5 + i) \]

Using the distributive property (FOIL method):
\[ P(x) = (x - 1)(x - 5) + (x - 1)(i) - (5i)(x - 5) - (5i)(i) \]

Calculating each part:
1. \( (x - 1)(x - 5) = x^2 - 6x + 5 \)
2. \( (x - 1)(i) = ix - i \)
3. \( - (5i)(x - 5) = -5ix + 25i \)
4. \( - (5i)(i) = -5(-1) = 5 \)

Combining these:
\[ P(x) = x^2 - 6x + 5 + ix - i - 5ix + 25i + 5 \]

### Step 4: Combine like terms
Now, combine the real and imaginary parts:
\[ P(x) = x^2 - 6x + 10 + (1 - 4i)x + 24i \]

### Step 5: Ensure rational coefficients
To ensure the polynomial has rational coefficients, we need to include the conjugate of the complex roots. The conjugate of \(1 + 5i\) is \(1 - 5i\) and the conjugate of \(5 - i\) is \(5 + i\).

Thus, we can also consider:
\[ P(x) = (x - (1 + 5i))(x - (1 - 5i))(x - (5 - i))(x - (5 + i)) \]

### Step 6: Expand the polynomial
Calculating the product of the conjugate pairs:
1. For \( (x - (1 + 5i))(x - (1 - 5i)) \):
\[ = (x - 1)^2 + (5i)^2 = (x - 1)^2 + 25 = x^2 - 2x + 26 \]
2. For \( (x - (5 - i))(x - (5 + i)) \):
\[ = (x - 5)^2 + i^2 = (x - 5)^2 + 1 = x^2 - 10x + 26 \]

### Step 7: Multiply the two results
Now, multiply the two quadratic polynomials:
\[ P(x) = (x^2 - 2x + 26)(x^2 - 10x + 26) \]

### Step 8: Expand the product
Using the distributive property:
\[ = x^4 - 10x^3 + 26x^2 - 2x^3 + 20x^2 - 52x + 26x^2 - 260x + 676 \]
Combining like terms:
\[ = x^4 - 12x^3 + 72x^2 - 312x + 676 \]

### Final Polynomial
Thus, the polynomial with rational coefficients whose roots are \(1 + 5i\) and \(5 - i\) is:
\[ P(x) = x^4 - 12x^3 + 72x^2 - 312x + 676 \]