given the function $c(x) = -(x - 5)(x + 1)(x - 3)$:
the coordinates of its $c$-intercept are
the coordinates of its $x$-intercepts are
the degree of $c$ is
as $x \to -\infty$, $f(x) \to$?
as $x \to \infty$, $f(x) \to$?
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Accepted Answer

### Part 1: C - intercept (y - intercept)
# Explanation:
## Step 1: Recall the definition of C - intercept
The C - intercept (or y - intercept) of a function $ C(x) $ occurs where $ x = 0 $. So we substitute $ x=0 $ into the function $ C(x)=-(x - 5)(x + 1)(x - 3) $.

## Step 2: Substitute $ x = 0 $ into the function
$$
\begin{align*}
C(0)&=-(0 - 5)(0 + 1)(0 - 3)\
&=-(-5)(1)(-3)\
&=- (15)\
&=- 15
\end{align*}
$$
So the coordinates of the C - intercept are $ (0,-15) $.

### Part 2: x - intercepts
# Explanation:
## Step 1: Recall the definition of x - intercepts
The x - intercepts of a function $ C(x) $ occur where $ C(x)=0 $. So we set $ -(x - 5)(x + 1)(x - 3)=0 $.

## Step 2: Solve for x
Since the product of factors is zero, then at least one of the factors must be zero.
- For $ x - 5=0 $, we get $ x = 5 $, so the point is $ (5,0) $.
- For $ x + 1=0 $, we get $ x=-1 $, so the point is $ (-1,0) $.
- For $ x - 3=0 $, we get $ x = 3 $, so the point is $ (3,0) $.

### Part 3: Degree of $ C(x) $
# Explanation:
## Step 1: Recall the degree of a polynomial
The degree of a polynomial is the highest power of $ x $ when the polynomial is expanded. The function $ C(x)=-(x - 5)(x + 1)(x - 3) $ is a product of three linear factors (degree 1 each). When we multiply three linear factors, the degree of the resulting polynomial is the sum of the degrees of the factors. Since each factor has degree 1, the degree of the product is $ 1 + 1+1=3 $.

### Part 4: End - behavior of the function
# Explanation:
## Step 1: Recall the leading term of the polynomial
First, we expand the polynomial (or find the leading term). The leading term of $ C(x)=-(x - 5)(x + 1)(x - 3) $ comes from multiplying the leading terms of each factor. The leading term of $ (x - 5) $ is $ x $, the leading term of $ (x + 1) $ is $ x $, the leading term of $ (x - 3) $ is $ x $, and we have a negative sign in front. So the leading term is $ -x\times x\times x=-x^{3} $.

## Step 2: Analyze the end - behavior based on the leading term
For a polynomial $ a_nx^n+\cdots+a_1x + a_0 $, the end - behavior is determined by the leading term $ a_nx^n $.
- If $ n $ is odd:
    - If $ a_n>0 $, as $ x
ightarrow\infty $, $ f(x)
ightarrow\infty $ and as $ x
ightarrow-\infty $, $ f(x)
ightarrow-\infty $.
    - If $ a_n<0 $, as $ x
ightarrow\infty $, $ f(x)
ightarrow-\infty $ and as $ x
ightarrow-\infty $, $ f(x)
ightarrow\infty $.

For $ C(x) $, the leading term is $ -x^{3} $ ( $ n = 3 $ which is odd and $ a_n=- 1<0 $).
- As $ x
ightarrow-\infty $, $ f(x)
ightarrow\infty $ (because for odd degree with negative leading coefficient, as $ x
ightarrow-\infty $, $ y
ightarrow\infty $).
- As $ x
ightarrow\infty $, $ f(x)
ightarrow-\infty $ (because for odd degree with negative leading coefficient, as $ x
ightarrow\infty $, $ y
ightarrow-\infty $).

# Answers:
- The coordinates of its $ C $ - intercept are $ \boldsymbol{(0,-15)} $.
- The coordinates of its $ x $ - intercepts are $ \boldsymbol{(5,0)}, \boldsymbol{(-1,0)}, \boldsymbol{(3,0)} $.
- The Degree of $ C $ is $ \boldsymbol{3} $.
- As $ x
ightarrow-\infty $, $ f(x)
ightarrow\boldsymbol{\infty} $.
- As $ x
ightarrow\infty $, $ f(x)
ightarrow\boldsymbol{-\infty} $.

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QUESTION IMAGE

Question

Explanation:

Part 1: C - intercept (y - intercept)

Step 1: Recall the definition of C - intercept

Step 2: Substitute \( x = 0 \) into the function

Part 2: x - intercepts

Step 1: Recall the definition of x - intercepts

Step 2: Solve for x

Part 3: Degree of \( C(x) \)

Step 1: Recall the degree of a polynomial

Part 4: End - behavior of the function

Answer: