a polynomial p is graphed. what could be the equation of p? choose 1 answer: a (p(x)=2x^{2}(x - 2)^{3}) b…

Explanation:

Step1: Analyze root - behavior

The graph touches the (x) - axis at (x = 0) and (x=2). If a polynomial (p(x)=a(x - r_1)^{n_1}(x - r_2)^{n_2}\cdots(x - r_k)^{n_k}), when (n_i) is even, the graph touches the (x) - axis at (x = r_i), and when (n_i) is odd, the graph crosses the (x) - axis at (x = r_i). The graph touches the (x) - axis at (x = 0) and (x = 2), so the exponents of (x) and ((x - 2)) in the polynomial equation should be even.

Step2: Check options

• Option A: (p(x)=2x^{2}(x - 2)^{3}), the exponent of ((x - 2)) is 3 (odd), so this option is incorrect.
• Option B: (p(x)=2x^{2}(x - 2)^{2}), the exponent of (x) is 2 and the exponent of ((x - 2)) is 2. This satisfies the condition that the graph touches the (x) - axis at (x = 0) and (x = 2).
• Option C: (p(x)=2x^{3}(x - 2)^{3}), the exponents of (x) and ((x - 2)) are 3 (odd), so this option is incorrect.
• Option D: (p(x)=2x^{3}(x - 2)^{2}), the exponent of (x) is 3 (odd), so this option is incorrect.

Answer:

B. (p(x)=2x^{2}(x - 2)^{2})