What is the additive inverse of the polynomial $-6x^3 + 4x^2 - 4x$?
To find the additive inverse of the polynomial \(-6x^3 + 4x^2 - 4x\), we need to identify a polynomial \(Q(x)\) such that the sum of the original polynomial and its inverse is zero. Mathematically, this means \(P(x) + Q(x) = 0\), which implies \(Q(x) = -P(x)\).
The given polynomial is \(P(x) = -6x^3 + 4x^2 - 4x\). To find its additive inverse \(Q(x)\), we must negate every term in the expression:
\[Q(x) = -(-6x^3 + 4x^2 - 4x)\]
Distributing the negative sign to each individual term, we get:
\[Q(x) = -(-6x^3) - (4x^2) - (-4x)\]
\[Q(x) = 6x^3 - 4x^2 + 4x\]
This matches option c from the list of choices provided.
The concept of an additive inverse is fundamental to solving equations. In everyday life, it is like balancing a bank account: if you spend money (a negative transaction), the additive inverse is earning an equal amount to return to a zero balance. Just as \(-5 + 5 = 0\), adding a polynomial to its additive inverse cancels it out, creating a balanced system of \(0\). This principle allows us to simplify complex algebraic expressions and isolate variables in various real-world engineering and financial calculations.