What is the product of all roots to the equation
\begin{align} & (x-1)(x-2)(x-3) + (x-2)(x-3)(x-4) \\ + & (x-3)(x-4)(x-5) + (x-4)(x-5)(x-6) \\ + & (x-5)(x-6)(x-7) + (x-6)(x-7)(x-8) =0 ? \end{align}
To find the product of all roots of the given equation, we need to find the values of "x" that satisfy the equation and then multiply them together.
Let's simplify the equation first:
\begin{align} & (x-1)(x-2)(x-3) + (x-2)(x-3)(x-4) \\ + & (x-3)(x-4)(x-5) + (x-4)(x-5)(x-6) \\ + & (x-5)(x-6)(x-7) + (x-6)(x-7)(x-8) = 0 \end{align}
Expanding each term and combining like terms, we get:
\begin{align} & (x^3 - 3x^2 + 2x)(x-3) + (x^3 - 6x^2 + 8x - 4)(x-4) \\ + & (x^3 - 9x^2 + 23x - 15)(x-5) + (x^3 - 12x^2 + 35x - 24)(x-6) \\ + & (x^3 - 15x^2 + 56x - 42)(x-7) + (x^3 - 18x^2 + 77x - 84)(x-8) = 0 \end{align}
Expanding further, we get:
\begin{align} & x^4 - 3x^3 - 3x^3 + 9x^2 + 2x^2 - 6x \\ - & 4x^3 + 12x^2 - 8x - 12x^2 + 36x - 24 \\ + & x^4 - 5x^3 - 9x^2 + 45x - 15x^3 + 75x^2 - 25x + 125 \\ - & 6x^4 + 30x^3 - 54x^2 + 324x - 35x^3 + 175x^2 - 245x + 420 \\ + & x^4 - 7x^3 + 15x^2 - 35x - 21x^3 + 147x^2 - 343x + 294 \\ - & 8x^4 + 56x^3 - 126x^2 + 392x - 28x^3 + 196x^2 - 392x + 672 = 0 \end{align}
Combining like terms, we obtain:
\begin{align} & -20x^4 + 114x^3 - 206x^2 + 430x - 20x^3 + 114x^2 - 230x + 420 = 0 \end{align}
Simplifying further, we get:
\begin{align} & -20x^4 + 94x^3 - 92x^2 + 200x + 420 = 0 \end{align}
Now, in order to find the roots of this equation, we can use numerical methods like factoring, quadratic formula, or graphing, as it doesn't appear to factor easily. Once we have the roots, we can multiply them together to obtain the product of all the roots.