What is the additive inverse of the polynomial $-6 x^3+4 x^2-4 x$?
The goal is to find the additive inverse of the polynomial \(P(x) = -6x^3 + 4x^2 - 4x\). By definition, the additive inverse of a polynomial \(P(x)\) is a polynomial \(Q(x)\) such that \(P(x) + Q(x) = 0\). This relationship implies that \(Q(x) = -P(x)\). To find this, we simply multiply the entire polynomial by \(-1\).
To find \(Q(x)\), we calculate the negative of the expression \(-6x^3 + 4x^2 - 4x\):
\[Q(x) = -(-6x^3 + 4x^2 - 4x)\]
Distributing the negative sign to each term inside the parentheses, we get:
\[Q(x) = -(-6x^3) - (4x^2) - (-4x)\]
\[Q(x) = 6x^3 - 4x^2 + 4x\]
This result represents the additive inverse of the original polynomial.
We now compare our result with the provided options:
Our result, \(6x^3 - 4x^2 + 4x\), matches option 'a'. Thus, the additive inverse is \(6x^3 - 4x^2 + 4x\). The final answer is \(\boxed{a}\).
In everyday financial contexts, the concept of an additive inverse is similar to balancing a ledger. If you spend \(\$50\) on an item (represented as \(-50\)), the additive inverse is the income of \(\$50\) (represented as \(+50\)), which brings your net change to \(0\). Similarly, when working with polynomials, identifying the additive inverse allows us to neutralize expressions, which is essential for simplifying complex algebraic equations in engineering or economic modeling.