Step 1: Verification of Assertion (A):
Given the polynomial \( p(x) = x^2 - 2x - 3 \). To find its zeroes, we solve \( x^2 - 2x - 3 = 0 \). Factoring the quadratic expression requires finding two numbers that multiply to \( -3 \) and add to \( -2 \). These numbers are \( -3 \) and \( 1 \), as \( (-3) \times 1 = -3 \) and \( (-3) + 1 = -2 \). Thus, the factored form is \( (x - 3)(x + 1) = 0 \). Setting each factor to zero yields the roots: \( x - 3 = 0 \Rightarrow x = 3 \) and \( x + 1 = 0 \Rightarrow x = -1 \). The zeroes of the polynomial are \( x = 3 \) and \( x = -1 \), confirming the assertion that the zeroes are \( -1 \) and \( 3 \).Step 2: Verification of Reason (R):
The graph of a quadratic polynomial intersects the x-axis at points where the polynomial's value is zero. These points are the x-intercepts, which correspond to the zeroes of the polynomial. As established in Step 1, the zeroes of \( p(x) = x^2 - 2x - 3 \) are \( x = -1 \) and \( x = 3 \). Therefore, the graph of this polynomial intersects the x-axis at \( (-1, 0) \) and \( (3, 0) \). This validates the reason that the graph intersects the x-axis at these specific points.Step 3: Conclusion:
Both the assertion and the reason are factually correct.