Q)Let a,b,c be rational numbers and f:Z-->Z be a function given by f(x)=ax² + bx+c.Then,a+b is

1. a negative integer
2. an integer
3. non-integral rational number
4. None

To determine the nature of \( a + b \) given the function \( f(x) = ax^2 + bx + c \) where \( a, b, c \) are rational numbers, we need to analyze the implications of the function's characteristics and the properties of rational numbers.

Understanding the Function

The function \( f(x) = ax^2 + bx + c \) is a quadratic function, which means it describes a parabola when graphed. The coefficients \( a \), \( b \), and \( c \) dictate the shape and position of this parabola. Specifically, the coefficient \( a \) determines whether the parabola opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).

Analyzing \( a + b \)

Now, let's focus on \( a + b \). Since both \( a \) and \( b \) are rational numbers, their sum \( a + b \) will also be a rational number. Rational numbers can be expressed as fractions where both the numerator and denominator are integers, and the denominator is not zero.

Possible Values of \( a + b \)

To classify \( a + b \) further, we need to consider the conditions under which it can be a negative integer, a non-integral rational number, or something else. Here are the possibilities:

• Negative Integer: If both \( a \) and \( b \) are negative, or if one is negative and the other is sufficiently small, \( a + b \) could indeed be a negative integer.
• Non-Integral Rational Number: If \( a \) and \( b \) are such that their sum does not simplify to an integer (for example, \( a = \frac{1}{2} \) and \( b = \frac{1}{3} \)), then \( a + b \) would be a non-integral rational number.
• Integer: If both \( a \) and \( b \) are integers, their sum will also be an integer.

Conclusion on the Nature of \( a + b \)

Without specific values for \( a \) and \( b \), we cannot definitively categorize \( a + b \) as a negative integer or a non-integral rational number. It could be any of these based on the values of \( a \) and \( b \). Therefore, the answer to the question is that \( a + b \) can be a negative integer, an integer, or a non-integral rational number depending on the specific values of \( a \) and \( b \).