The given linear polynomial y = f(x) has(a) 2 zeros(b) 1 zero and the zero is ‘3’(c) 1 zero and the zero is ‘4’(d) No zero1

To determine the characteristics of the linear polynomial $y = f(x)$, we need to understand that a linear polynomial generally takes the form:

$f(x) = ax + b$

where $a$ and $b$ are constants and $a \neq 0$. The number of zeros (or roots) of a polynomial is the number of times the polynomial equals zero, i.e., solving $f(x) = 0$.

Analysis of Each Option

1. (a) 2 zeros

   • A linear polynomial can only have a maximum of 1 zero because it is represented by a straight line which can intersect the x-axis at most once.

2. (b) 1 zero and the zero is ‘3’

   • This scenario is valid. For the polynomial to have a zero at $x = 3$: $f(3) = a(3) + b = 0 \Rightarrow b = -3a$
   • Thus, there exists a linear polynomial with a zero at $x = 3$.

3. (c) 1 zero and the zero is ‘4’

   • Similar to option (b), this is also valid. For the polynomial to have a zero at $x = 4$: $f(4) = a(4) + b = 0 \Rightarrow b = -4a$
   • Therefore, this also represents a valid linear polynomial.

4. (d) No zero

   • A linear polynomial will have no zeros only if $b \neq 0$ and the linear expression does not cross the x-axis. However, this is not possible unless $f(x)$ is a constant function, but that does not fit the definition of a linear polynomial.

Conclusion

Based on this analysis:

• Options (a) is incorrect.
• Options (b) and (c) are correct possible scenarios for a linear polynomial.
• Option (d) is incorrect.

Final Answer

The linear polynomial $y = f(x)$ can have:

• (b) 1 zero at $x = 3$
• (c) 1 zero at $x = 4$

Thus, options (b) and (c) are valid possibilities for the zeros of the polynomial.