We can use the right outside diagonal of the empty Calpas's triangle as the coefficients of a polynomial: t^3 + 6t^2 + 12t + 8. If we replace every t with t-1, we get (t-1)^3 + 6(t-1)^2 + 12(t-1) + 8. Expand and simplify this polynomial. Enter the polynomial as your answer.
+310 Let's break it down and simplify each part.
\((t-1)^3\)= \(t^3-3t^2+3t-1\)
\(6(t-1)^2\)\( = 6(t^2-2t+1) = 6t^2-12t+6\)
\(12(t-1)=12t-12\)
Combining these and the 8 together, we get our final answer: \(t^3-3t^2+3t-1\) \( + 6t^2-12t+6\) \( + 12t-12+8\) = \(t^3+3t^2-3t+1\)
