Solve (x-1)(x+1)(x-2) = -2x²+x³: Comparing Polynomial Forms

Solve the following equation:

$(x-1)(x+1)(x-2)=-2x^2+x^3$

Let's examine the given equation:

$(x-1)(x+1)(x-2)=-2x^2+x^3$First, let's simplify the equation, using the perfect square difference formula:

$(a+b)(a-b)=a^2-b^2$and the expanded distribution law,

We'll start by opening the first pair of parentheses from the left which is in the left side using the perfect square difference formula mentioned, we'll put the result in new parentheses (since the entire expression is multiplied by the expression in the unopened parentheses) then we'll simplify the expression in the parentheses:

$(x-1)(x+1)(x-2)=-2x^2+x^3 \\ \downarrow\\ \textcolor{blue}{(}x^2-1^2\textcolor{blue}{)}(x-2)=-2x^2+x^3\\ \textcolor{blue}{(}x^2-1\textcolor{blue}{)}(x-2)=-2x^2+x^3\\$We'll continue using the expanded distribution law and open the parentheses on the left side, then we'll move terms and combine like terms:

$(x^2-1)(x-2)=-2x^2+x^3\\ \downarrow\\ x^3-2x^2-x+2=-2x^2+x^3\\ x^3-2x^2-x+2+2x^2-x^3=0\\ -x+2=0\\ -x=-2\hspace{6pt}\text{/}\cdot(-1)\\ \boxed{x=2}$

Let's summarize the equation solving steps:

$(x-1)(x+1)(x-2)=-2x^2+x^3 \\ \downarrow\\ \textcolor{blue}{(}x^2-1\textcolor{blue}{)}(x-2)=-2x^2+x^3\\ \downarrow\\ x^3-2x^2-x+2=-2x^2+x^3\\ \boxed{x=2}$Therefore, the correct answer is answer C.