The quadratic polynomial `p(x)` ha following properties `p(x)` can be positive or zero for all real numbers `p(1)=0a n dp(2)=2.` Then find the quadratic polynomial.

Explanation

To find the quadratic polynomial \( p(x) \) that satisfies the given properties, we can follow these steps:

### Step 1: Understanding the properties of the polynomial
The polynomial \( p(x) \) must be non-negative for all real numbers. This suggests that \( p(x) \) can be expressed in the form of a square of a linear polynomial, since squares are always non-negative.

### Step 2: Formulating the polynomial
Given that \( p(x) \) can be expressed as a square, we can write:
\[ p(x) = k(x - 1)^2 \]
where \( k \) is a positive constant. This form ensures that \( p(x) \) is non-negative for all \( x \) and equals zero when \( x = 1 \).

### Step 3: Using the second condition
We know from the problem that \( p(1) = 0 \). Substituting \( x = 1 \) into our polynomial:
\[ p(1) = k(1 - 1)^2 = k(0) = 0 \]
This condition is satisfied for any \( k \).

### Step 4: Using the third condition
Next, we use the condition \( p(2) = 2 \). Substituting \( x = 2 \) into our polynomial:
\[ p(2) = k(2 - 1)^2 = k(1)^2 = k \]
According to the condition, this must equal 2:
\[ k = 2 \]

### Step 5: Writing the final polynomial
Now that we have determined \( k \), we can substitute it back into our polynomial expression:
\[ p(x) = 2(x - 1)^2 \]

### Step 6: Expanding the polynomial (optional)
If required, we can expand this polynomial:
\[ p(x) = 2(x^2 - 2x + 1) = 2x^2 - 4x + 2 \]

Thus, the quadratic polynomial \( p(x) \) is:
\[ \boxed{2(x - 1)^2} \quad \text{or} \quad \boxed{2x^2 - 4x + 2} \]