Question

foundations & fundamentals
these problems are meant to be straightforward and focus on basic concepts.
sketch the graph of the polynomial function. credit for graphing polynomial functions is based on correctness, cleanliness, proper end - behavior, illustration of proper behavior near the zeros, and the labeling of all x - intercepts.
$p(x)=-3.3(x - 2)(x + 4)^{3}(x + 10)^{5}$
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factor p

Explanation:

Step1: Identify the factors

The polynomial $P(x)= - 3.3(x - 2)(x + 4)^{3}(x + 10)^{5}$ is already factored. The factors are $-3.3$, $(x - 2)$, $(x + 4)^{3}$ and $(x + 10)^{5}$.

Step2: Analyze the x - intercepts

Set each non - constant factor equal to zero.
For $x-2=0$, we get $x = 2$.
For $(x + 4)^{3}=0$, we get $x=-4$.
For $(x + 10)^{5}=0$, we get $x=-10$.

Step3: Analyze the end - behavior

The leading term of the polynomial is found by multiplying the leading terms of each factor. The degree of the polynomial is $1+3 + 5=9$ (an odd number) and the leading coefficient is $-3.3$ (negative). As $x\to+\infty$, $y\to-\infty$ and as $x\to-\infty$, $y\to+\infty$.

Step4: Analyze the behavior near the zeros

• Near $x = 2$: Since the factor $(x - 2)$ has a power of 1, the graph crosses the x - axis at $x = 2$.
• Near $x=-4$: Since the factor $(x + 4)$ has a power of 3 (an odd number), the graph crosses the x - axis at $x=-4$.
• Near $x=-10$: Since the factor $(x + 10)$ has a power of 5 (an odd number), the graph crosses the x - axis at $x=-10$.

To sketch the graph:

1. Plot the x - intercepts at $x = 2$, $x=-4$ and $x=-10$.
2. Use the end - behavior to draw the graph going down as $x\to+\infty$ and up as $x\to-\infty$.
3. Show the graph crossing the x - axis at each of the x - intercepts.

Answer:

The polynomial $P(x)$ is already factored as $P(x)= - 3.3(x - 2)(x + 4)^{3}(x + 10)^{5}$. The x - intercepts are $x = 2$, $x=-4$ and $x=-10$. The end - behavior is that as $x\to+\infty$, $y\to-\infty$ and as $x\to-\infty$, $y\to+\infty$. The graph crosses the x - axis at each of the x - intercepts.