$p = (w-30)(w^2+178w+7921)$ Given that (-89\)\ is a double zero of the polynomial equation, which of the following could be the graph of the equation in the $w$ - $p$ ‑plane?

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p = (w-30)(w^2+178w+7921)

Given that (-89\)\ is a double zero of the polynomial equation, which of the following could be the graph of the equation in the

w

p

‑plane?

Given Polynomial Analysis: The given polynomial is $p = (w-30)(w^2+178w+7921)$ . We are told that $-89$ is a double zero of the polynomial. This means that $(w + 89)$ is a factor of the polynomial, and since it is a double zero, $(w + 89)^2$ should be a factor.
Factor Verification: Let's check if $(w + 89)^2$ is indeed a factor of the second part of the polynomial, which is $w^2 + 178w + 7921$ . We can expand $(w + 89)^2$ to see if it matches. $\newline$ $(w + 89)^2 = w^2 + 2\cdot89\cdot w + 89^2 = w^2 + 178w + 7921$ .
Confirmation of Double Zero: Since $(w + 89)^2$ expands to $w^2 + 178w + 7921$ , we can confirm that $-89$ is indeed a double zero of the polynomial. This means that the polynomial can be rewritten as $p = (w-30)(w + 89)^2$ .
Graph Consideration: Now, we need to consider the graph of the polynomial in the $w$ -plane. The graph will touch the $w$ -axis at $w = -89$ and cross the $w$ -axis at $w = 30$ , because these are the zeros of the polynomial. Since $-89$ is a double zero, the graph will touch the $w$ -axis at $w = -89$ and bounce off, indicating a change in concavity but not crossing the axis.
Leading Coefficient Analysis: The leading coefficient of the polynomial is positive, as we can see from the expanded form of the polynomial. This means that the ends of the graph will go up as $w$ approaches positive and negative infinity.
Graph Behavior Prediction: Given the information about the zeros and the leading coefficient, the graph of the polynomial will start from the upper left, approach the w-axis near $w = -89$ , touch and bounce off at $w = -89$ , then decrease until it crosses the w-axis at $w = 30$ , and finally go up towards the upper right.