ASSERTION (A): The zeroes of the polynomial f(x) = x ^ 2 - 5x + 6 are REASON (R): A linear polynomial has exactly one

In this question, we need to analyze both the **Assertion (A)** and the **Reason (R)** individually, and then determine if the Reason is a valid explanation of the Assertion.

### **Assertion (A):**
"The zeroes of the polynomial \( f(x) = x^2 - 5x + 6 \) are..."

This is a quadratic equation. To find the zeroes (solutions) of \( f(x) = x^2 - 5x + 6 \), we can factorize it:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
Thus, the zeroes of the polynomial are \( x = 2 \) and \( x = 3 \).

### **Reason (R):**
"A linear polynomial has exactly one zero."

This statement is true. A linear polynomial of the form \( ax + b = 0 \) has exactly one solution (or zero). For example, if \( f(x) = 2x + 3 \), it will have one zero when \( 2x + 3 = 0 \), giving \( x = -\frac{3}{2} \).

### **Analyzing the Assertion and Reason together:**
- **Assertion (A)** is true: The quadratic polynomial \( f(x) = x^2 - 5x + 6 \) has two zeroes: \( x = 2 \) and \( x = 3 \).
- **Reason (R)** is also true: A linear polynomial does indeed have exactly one zero.
- **However, the Reason does not explain the Assertion** because the Assertion is about a quadratic polynomial, while the Reason talks about a linear polynomial. The two statements are unrelated in this context.

### Conclusion:
- **Assertion (A) is true.**
- **Reason (R) is true, but it is not the correct explanation of the Assertion (A).**

Would you like further details on solving quadratic equations, or do you have any questions?

Here are 5 related questions for deeper understanding:
1. How do you factorize quadratic polynomials like \( x^2 - 5x + 6 \)?
2. Can the quadratic formula be used to solve \( f(x) = x^2 - 5x + 6 \) instead of factorization?
3. What happens if a quadratic equation has no real roots?
4. How can we graphically interpret the zeroes of a quadratic polynomial?
5. How do we derive the number of zeroes of a polynomial based on its degree?

**Tip:** A quadratic polynomial can have two real, one real (repeated), or no real roots depending on its discriminant \( b^2 - 4ac \).

ASSERTION (A): The zeroes of the polynomial f(x) = x^2 - 5x + 6 are REASON (R): A linear polynomial has exactly one

Learn how to solve the quadratic equation x^2 - 5x + 6 and understand the distinction between linear and quadratic polynomials. This problem includes key algebraic concepts such as factoring and analyzing polynomial roots, and is suitable for students in Grades 7-9.

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