How many zeroes will be there for the polynomial `f(x) = (x - 2)^2 + 4` ?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find how many zeroes the polynomial \( f(x) = (x - 2)^2 + 4 \) has, we can follow these steps:

### Step 1: Analyze the polynomial
The polynomial is given as \( f(x) = (x - 2)^2 + 4 \). This is a quadratic polynomial in standard form.

### Step 2: Determine the minimum value
The term \( (x - 2)^2 \) is always non-negative (it is a square term), and its minimum value occurs when \( x = 2 \). At this point, \( (x - 2)^2 = 0 \).

### Step 3: Calculate \( f(2) \)
Now, substituting \( x = 2 \) into the polynomial:
\[ f(2) = (2 - 2)^2 + 4 = 0 + 4 = 4 \]
Thus, the minimum value of \( f(x) \) is 4.

### Step 4: Analyze the implications
Since the minimum value of \( f(x) \) is 4, and the polynomial does not go below this value, it means that \( f(x) \) cannot equal zero for any real value of \( x \).

### Step 5: Conclusion
Therefore, the polynomial \( f(x) = (x - 2)^2 + 4 \) has no real zeroes.

### Final Answer
The number of zeroes for the polynomial \( f(x) = (x - 2)^2 + 4 \) is **0**.

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