The function f is defined by f(x) = 2x³ + 3x² + cx + 8, where c is a constant. In the xy-plane, the graph of f intersects the x-axis at the three points (−4, 0), (1/2, 0), and (p, 0). What is the value of c?

Given that the function \( f(x) = 2x^3 + 3x^2 + cx + 8 \) intersects the x-axis at the points \( (-4, 0) \), \( \left(\frac{1}{2}, 0\right) \), and \( (p, 0) \), it implies that these values are the roots of the equation \( f(x) = 0 \).

The function can be expressed in factored form based on the given roots:

\[
f(x) = 2(x + 4)\left(x - \frac{1}{2}\right)(x - p)
\]

To determine the value of \( c \), we will expand this factored form and then compare the coefficients with the original function.

### Step 1: Expand the factored form
Let's first expand \( (x + 4)(x - \frac{1}{2}) \):

\[
(x + 4)\left(x - \frac{1}{2}\right) = x^2 - \frac{1}{2}x + 4x - 2 = x^2 + \frac{7}{2}x - 2
\]

Now multiply by \( (x - p) \):

\[
2(x^2 + \frac{7}{2}x - 2)(x - p)
\]

Distribute \( (x - p) \):

\[
= 2\left[x^3 - px^2 + \frac{7}{2}x^2 - \frac{7p}{2}x - 2x + 2p\right]
\]

Now, let's collect the like terms:

\[
= 2x^3 + \left(7 - 2p\right)x^2 + \left(\frac{7}{2} - 7p\right)x + 4p
\]

### Step 2: Compare with the original function

The original function is:

\[
f(x) = 2x^3 + 3x^2 + cx + 8
\]

Comparing coefficients with the expanded form:

- Coefficient of \( x^2 \): \( 7 - 2p = 3 \)
- Coefficient of \( x \): \( \frac{7}{2} - 7p = c \)
- Constant term: \( 4p = 8 \)

### Step 3: Solve for \( p \) and \( c \)

First, solve for \( p \) from the constant term:

\[
4p = 8 \implies p = 2
\]

Now substitute \( p = 2 \) into \( 7 - 2p = 3 \):

\[
7 - 2(2) = 3
\]

This confirms that \( p = 2 \) is correct.

Now substitute \( p = 2 \) into the expression for \( c \):

\[
c = \frac{7}{2} - 7(2)
\]

\[
c = \frac{7}{2} - 14 = \frac{7}{2} - \frac{28}{2} = \frac{-21}{2} = -10.5
\]

Thus, the value of \( c \) is \( \boxed{-\frac{21}{2}} \) or \( -10.5 \).

Would you like further details or have any other questions?

Here are 5 related questions to consider:
1. How would the roots change if the value of \( c \) were different?
2. Can you find the sum of the roots for the cubic equation?
3. What is the relationship between the roots and coefficients in a cubic equation?
4. How do the points of intersection affect the behavior of the function?
5. How would you determine the value of \( p \) if \( c \) were known?

**Tip:** When comparing coefficients of polynomials, ensure that each corresponding term is correctly matched to solve for unknowns.

Explore how to find the value of c in the polynomial function f(x) = 2x^3 + 3x^2 + cx + 8 using given roots (-4, 0), (1/2, 0), and (p, 0). This problem involves polynomial functions, root determination, and coefficient comparison techniques. Suitable for advanced high school students.

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