The function g is defined as g(x) = x^3 + (2 - a)x^2 - 2ax, where a is a constant. When the value of x is either 0, -2, or 8, the value of g(x) is 0. What is the value of a?

To determine the value of \( a \) for which \( g(x) = 0 \) when \( x = 0, -2, \) and \( 8 \), we begin by evaluating \( g(x) \) at these points.

The function \( g \) is given by:
\[ g(x) = x^3 + (2 - a)x^2 - 2ax \]

First, evaluate \( g(x) \) at \( x = 0 \):
\[ g(0) = 0^3 + (2 - a)0^2 - 2a \cdot 0 = 0 \]

Since \( g(0) = 0 \), it does not provide any additional information about \( a \).

Next, evaluate \( g(x) \) at \( x = -2 \):
\[ g(-2) = (-2)^3 + (2 - a)(-2)^2 - 2a(-2) \]
\[ g(-2) = -8 + (2 - a)4 + 4a \]
\[ g(-2) = -8 + 8 - 4a + 4a \]
\[ g(-2) = -8 + 8 = 0 \]

This simplifies to \( 0 = 0 \), giving no new information about \( a \).

Finally, evaluate \( g(x) \) at \( x = 8 \):
\[ g(8) = 8^3 + (2 - a)8^2 - 2a \cdot 8 \]
\[ g(8) = 512 + (2 - a)64 - 16a \]
\[ g(8) = 512 + 128 - 64a - 16a \]
\[ g(8) = 512 + 128 - 80a \]
\[ g(8) = 640 - 80a \]

Since \( g(8) = 0 \):
\[ 640 - 80a = 0 \]
\[ 80a = 640 \]
\[ a = \frac{640}{80} \]
\[ a = 8 \]

Thus, the value of \( a \) is \( 8 \).

Would you like more details or have any questions? Here are 5 questions you might ask next:
1. How do you determine the roots of a polynomial equation?
2. Can you explain how to factorize cubic polynomials?
3. What are the properties of polynomial functions?
4. How can the value of \( a \) affect the shape of the graph of \( g(x) \)?
5. What methods can be used to verify the roots of a polynomial equation?

**Tip:** Always check the given conditions in a problem to see if they provide independent equations, which can simplify solving for unknowns.

Discover how to determine the value of a constant 'a' in the polynomial function g(x) = x^3 + (2 - a)x^2 - 2ax when given specific conditions. This problem involves evaluating the function at multiple points and solving for the constant 'a'. Suitable for high school students in Grades 10-12.

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