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.