Question

find the zeros of the polynomial function below.

\\p(x) = (x + 6)(x + 7)(x - 2)^2(x + 1)^3(x - 3)\\

plot all the zeros (x-intercepts) of the polynomial in the interactive graph below

Explanation:

⚡ Using what you learned: Zeros and Multiplicity

Step 1: Set the polynomial to zero

To find the zeros of the polynomial function \( p(x) \), set \( p(x) = 0 \):

\[ (x + 6)(x + 7)(x - 2)^2(x + 1)^3(x - 3) = 0 \]

Step 2: Solve for each factor

Set each distinct linear factor to zero:

• \( x + 6 = 0 \implies x = -6 \)
• \( x + 7 = 0 \implies x = -7 \)
• \( x - 2 = 0 \implies x = 2 \)
• \( x + 1 = 0 \implies x = -1 \)
• \( x - 3 = 0 \implies x = 3 \)

Step 3: Identify the coordinates to plot

The zeros correspond to the \( x \)-intercepts on the graph, which are points where the graph crosses or touches the \( x \)-axis:

\[ (-7, 0), (-6, 0), (-1, 0), (2, 0), (3, 0) \]

Answer:

The zeros of the polynomial function are \( x = -7, -6, -1, 2, 3 \).

To complete the interactive graph, plot points at the following coordinates on the \( x \)-axis:

• \( (-7, 0) \)
• \( (-6, 0) \)
• \( (-1, 0) \)
• \( (2, 0) \)
• \( (3, 0) \)