In the standard form of quadratic polynomial, `Ax^2+Bx+C,A,B and C:`

Options

A. all are real numbers
B. all are rational numbers
C. a' is a non-zero real number and B and C are any real numbers
D. all are integers

Answer

To solve the question regarding the standard form of a quadratic polynomial, we need to analyze the form given and the conditions for the coefficients \(A\), \(B\), and \(C\).

### Step-by-Step Solution:

1. **Understanding the Standard Form**:
The standard form of a quadratic polynomial is given as:
\[ Ax^2 + Bx + C \]
where \(A\), \(B\), and \(C\) are coefficients.

**Hint**: Remember that the standard form is characterized by the presence of the \(x^2\) term.

2. **Identifying the Coefficient \(A\)**:
For a polynomial to be classified as a quadratic polynomial, the coefficient \(A\) (the coefficient of \(x^2\)) must be non-zero. If \(A = 0\), the polynomial would no longer be quadratic but linear.

**Hint**: Think about what happens to the polynomial if the \(x^2\) term is absent.

3. **Values of Coefficients**:
The coefficients \(A\), \(B\), and \(C\) can be any real numbers, but \(A\) specifically must not be zero. This means:
- \(A\) can be any non-zero real number.
- \(B\) and \(C\) can be any real numbers (including zero).

**Hint**: Consider the implications of \(A\) being zero versus being a non-zero value.

4. **Conclusion**:
Therefore, the correct interpretation of the values of \(A\), \(B\), and \(C\) in the context of the standard form of a quadratic polynomial is:
- \(A\) is a non-zero real number.
- \(B\) and \(C\) can be any real numbers.

Thus, the correct answer is that \(A\) must be a non-zero real number, while \(B\) and \(C\) can be any real number.

### Final Answer:
The correct condition for the coefficients in the standard form of a quadratic polynomial \(Ax^2 + Bx + C\) is:
- \(A\) is a non-zero real number.
- \(B\) and \(C\) can be any real numbers.