Assertion (A): Zeroes of a polynomial \(p(x) = x^2 − 2x − 3\) are -1 and 3.
Reason (R): The graph of polynomial \(p(x) = x^2 − 2x − 3\) intersects the x-axis at (-1, 0) and (3, 0).
Reason (R): The graph of polynomial \(p(x) = x^2 − 2x − 3\) intersects the x-axis at (-1, 0) and (3, 0).
- Both, Assertion (A) and Reason (R) are true. Reason (R) explains Assertion (A) completely.
- Both, Assertion (A) and Reason (R) are true. Reason (R) does not explain Assertion (A).
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
The Correct Option is A
Solution and Explanation
- The given polynomial is \(p(x) = x^2 - 2x - 3\). To find the zeroes, we solve \(x^2 - 2x - 3 = 0\) by factoring:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 \]
Thus, the zeroes of the polynomial are \(x = 3\) and \(x = -1\).
- The graph of a quadratic polynomial intersects the \(x\)-axis at its zeroes. Therefore, the points where the graph intersects the \(x\)-axis are \((-1, 0)\) and \((3, 0)\), as given in the reason.
Since both the assertion and the reason are true, and the reason explains the assertion, the correct answer is (a).
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