What is the additive inverse of the polynomial?
* [tex]$-6 x^3+4 x^2-4 x$[/tex]
* [tex]$6 x^3+4 x^2+4 x$[/tex]
* [tex]$6 x^3-4 x^2+4 x$[/tex]
* [tex]$-6 x^3-4 x^2-4 x$[/tex]
* [tex]$6 x^3+4 x^2-4 x$[/tex]
Answer (1)
To find the additive inverse of a polynomial, change the sign of each term.
Multiply the polynomial by − 1 : − 1 × ( − 6 x 3 + 4 x 2 − 4 x ) .
Distribute the − 1 to each term: 6 x 3 − 4 x 2 + 4 x .
The additive inverse is 6 x 3 − 4 x 2 + 4 x .
Explanation
Understanding the Problem The problem asks us to find the additive inverse of the polynomial − 6 x 3 + 4 x 2 − 4 x . The additive inverse is the polynomial that, when added to the original polynomial, results in zero. To find it, we change the sign of each term in the polynomial.
Finding the Additive Inverse To find the additive inverse, we multiply the entire polynomial by − 1 :
− 1 × ( − 6 x 3 + 4 x 2 − 4 x ) Distribute the − 1 to each term: ( − 1 ) × ( − 6 x 3 ) + ( − 1 ) × ( 4 x 2 ) + ( − 1 ) × ( − 4 x ) This simplifies to: 6 x 3 − 4 x 2 + 4 x
Final Answer Therefore, the additive inverse of the polynomial − 6 x 3 + 4 x 2 − 4 x is 6 x 3 − 4 x 2 + 4 x .
Examples
Additive inverses are useful in many areas of math. For example, when solving equations, we often add the additive inverse to both sides to isolate a variable. If you have the equation x + 5 = 10 , you can add the additive inverse of 5, which is -5, to both sides to get x + 5 − 5 = 10 − 5 , which simplifies to x = 5 . This concept is also used in physics, such as when calculating net forces, where forces in opposite directions can cancel each other out, acting as additive inverses.
