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.