8(x-a)(x-b)(x-c) where a,b and c were constants and (r,s) is a solution.

Given the expression \( 8(x-a)(x-b)(x-c) \), where \( a \), \( b \), and \( c \) are constants, and \( (r, s) \) is a solution, we need to clarify whether \( r \) and \( s \) are roots of the equation formed by setting this expression equal to zero or if they are specific values for \( x \) and the corresponding value of the expression.

### Case 1: \( r \) and \( s \) as roots
If \( (r, s) \) represents roots of the polynomial equation \( 8(x-a)(x-b)(x-c) = 0 \), then \( r \), \( s \), and another value (since it's a cubic polynomial) are the roots of the equation. So,

\[
r = a, \quad s = b \quad \text{or} \quad r = a, \quad s = c \quad \text{or} \quad r = b, \quad s = c
\]

This would mean that \( r \), \( s \), and the third constant (which is \( a \), \( b \), or \( c \) depending on the pairs chosen) satisfy the equation individually.

### Case 2: \( (r, s) \) as an \( x \)-value and corresponding polynomial value
If \( r \) is the value of \( x \), and \( s \) is the value of the polynomial at \( x = r \), then:

\[
s = 8(r-a)(r-b)(r-c)
\]

In this case, \( r \) is not necessarily equal to \( a \), \( b \), or \( c \); it's just the input for \( x \) in the polynomial, and \( s \) is the corresponding output.

Would you like to explore one of these scenarios in detail or discuss a different interpretation?

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Here are 5 related questions that expand on this information:

1. If \( r \) and \( s \) are roots, what would the third root of the polynomial be?
2. How would the expression change if \( a \), \( b \), and \( c \) are complex numbers?
3. What would the polynomial look like if we expanded \( 8(x-a)(x-b)(x-c) \)?
4. How can we find the vertex of the curve defined by \( 8(x-a)(x-b)(x-c) = 0 \)?
5. What is the significance of the constant 8 in the expression?

**Tip:** When analyzing polynomials, carefully consider whether you are dealing with roots or specific function values, as this affects the interpretation and solution method.

Explore the polynomial expression 8(x-a)(x-b)(x-c) and understand its solutions (r, s). This problem examines different scenarios where (r, s) could represent roots or specific values in the context of cubic polynomials. Suitable for advanced high school students.

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