Question

question
written as a simplified polynomial in standard form, what is the result when $(2t - 1)^2$ is subtracted from $t^3$?
answer

Explanation:

Step1: Expand \((2t - 1)^3\)

Using the formula \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\), where \(a = 2t\) and \(b = 1\).
\[

$$\begin{align*} (2t - 1)^3&=(2t)^3 - 3(2t)^2(1) + 3(2t)(1)^2 - 1^3\\ &=8t^3 - 12t^2 + 6t - 1 \end{align*}$$

\]

Step2: Subtract \((2t - 1)^3\) from \(t^3\)

The expression is \(t^3 - (8t^3 - 12t^2 + 6t - 1)\).
\[

$$\begin{align*} t^3 - 8t^3 + 12t^2 - 6t + 1&=(1 - 8)t^3 + 12t^2 - 6t + 1\\ &=-7t^3 + 12t^2 - 6t + 1 \end{align*}$$

\]

Answer:

\(-7t^3 + 12t^2 - 6t + 1\)