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Answer :
Sure, let's find the additive inverse of the given polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex].
### Step-by-Step Solution:
1. Understand the Concept of Additive Inverse:
- The additive inverse of a polynomial is another polynomial that, when added to the original, results in zero. Essentially, you change the sign of each term in the polynomial.
2. Change the Sign of Each Term:
- For the polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex]:
- The term [tex]\(9x^3\)[/tex] becomes [tex]\(-9x^3\)[/tex].
- The term [tex]\(-7x^2\)[/tex] becomes [tex]\(+7x^2\)[/tex].
- The term [tex]\(-13x\)[/tex] becomes [tex]\(+13x\)[/tex].
- The constant term [tex]\(+2\)[/tex] becomes [tex]\(-2\)[/tex].
3. Write the Additive Inverse Polynomial:
- Combining all these changed terms together, we get:
[tex]\[
-9x^3 + 7x^2 + 13x - 2
\][/tex]
### Conclusion:
The additive inverse of the polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex] is:
[tex]\[
-9x^3 + 7x^2 + 13x - 2
\][/tex]
This corresponds to option (3) [tex]\(-9x^3 - 7x^2 - 13x - 2\)[/tex].
### Step-by-Step Solution:
1. Understand the Concept of Additive Inverse:
- The additive inverse of a polynomial is another polynomial that, when added to the original, results in zero. Essentially, you change the sign of each term in the polynomial.
2. Change the Sign of Each Term:
- For the polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex]:
- The term [tex]\(9x^3\)[/tex] becomes [tex]\(-9x^3\)[/tex].
- The term [tex]\(-7x^2\)[/tex] becomes [tex]\(+7x^2\)[/tex].
- The term [tex]\(-13x\)[/tex] becomes [tex]\(+13x\)[/tex].
- The constant term [tex]\(+2\)[/tex] becomes [tex]\(-2\)[/tex].
3. Write the Additive Inverse Polynomial:
- Combining all these changed terms together, we get:
[tex]\[
-9x^3 + 7x^2 + 13x - 2
\][/tex]
### Conclusion:
The additive inverse of the polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex] is:
[tex]\[
-9x^3 + 7x^2 + 13x - 2
\][/tex]
This corresponds to option (3) [tex]\(-9x^3 - 7x^2 - 13x - 2\)[/tex].
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