If ` f(x) ` is a polynomial of degree 4 with rational coefficients
and touches x - axis at ` (sqrt(2) , 0 )` , then for the equation
` f(x) = 0`,

Explanation

To solve the problem, we need to find the polynomial \( f(x) \) of degree 4 that touches the x-axis at the point \( (\sqrt{2}, 0) \). Since it touches the x-axis, this means that \( \sqrt{2} \) is a double root of the polynomial.

### Step-by-Step Solution:

1. **Identify the roots**: Since the polynomial \( f(x) \) touches the x-axis at \( (\sqrt{2}, 0) \), it means that \( \sqrt{2} \) is a root with even multiplicity. Therefore, we can express this as:
\[ f(x) = k(x - \sqrt{2})^2(x - r_1)(x - r_2) \]
...