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------------------------------------------------ What is the additive inverse of [tex]$4x^3 - 9x^2 + x - 7$[/tex]?

A. [tex]$-4x^3 + 9x^2 - x - 7$[/tex]
B. [tex]$-4x^3 + 9x^2 + x - 7$[/tex]
C. [tex]$-4x^3 + 9x^2 - x + 7$[/tex]
D. [tex]$-4x^3 - 9x^2 - x - 7$[/tex]

To find the additive inverse of a polynomial, you simply change the sign of each term in the polynomial. Let's work through the polynomial [tex]$4x^3 - 9x^2 + x - 7$[/tex] together.

1. Start with the polynomial:
[tex]$4x^3 - 9x^2 + x - 7$[/tex]

2. Change the sign of each term:
- The term [tex]$4x^3$[/tex] becomes [tex]$-4x^3$[/tex].
- The term [tex]$-9x^2$[/tex] becomes [tex]$+9x^2$[/tex].
- The term [tex]$+x$[/tex] becomes [tex]$-x$[/tex].
- The term [tex]$-7$[/tex] becomes [tex]$+7$[/tex].

3. Combine the negated terms to get the additive inverse:
[tex]$-4x^3 + 9x^2 - x + 7$[/tex]

So, the additive inverse of the polynomial [tex]$4x^3-9x^2+x-7$[/tex] is [tex]$-4x^3 + 9x^2 - x + 7$[/tex]. Looking at your answer choices, this matches option C.