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Answer :
Sure, let's solve the problem step-by-step.
We need to find the additive inverse of the polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex].
Step 1: Understand what the additive inverse means.
The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, yields the zero polynomial. Essentially, it is obtained by negating each coefficient in the original polynomial.
Step 2: Negate each term in 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 term [tex]\(+2\)[/tex] becomes [tex]\(-2\)[/tex].
Step 3: Write the additive inverse using these new terms.
So, the additive inverse of [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex] is:
[tex]\[
-9x^3 + 7x^2 + 13x - 2
\][/tex]
Step 4: Compare with the given options.
The correct answer from the options provided should match this polynomial:
1) [tex]\(9x^2 + 7x^2 + 13x - 2\)[/tex] - Incorrect (wrong polynomial structure)
2) [tex]\(-9x^3 + 7x^2 + 13x - 2\)[/tex] - Correct
3. Option 3 is missing, but let's proceed.
4) [tex]\(9x^3 + 7x^2 - 13x - 2\)[/tex] - Incorrect (not the additive inverse)
Thus, the correct answer is:
[tex]\[
\boxed{-9x^3 + 7x^2 + 13x - 2}
\][/tex]
We need to find the additive inverse of the polynomial [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex].
Step 1: Understand what the additive inverse means.
The additive inverse of a polynomial is a polynomial that, when added to the original polynomial, yields the zero polynomial. Essentially, it is obtained by negating each coefficient in the original polynomial.
Step 2: Negate each term in 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 term [tex]\(+2\)[/tex] becomes [tex]\(-2\)[/tex].
Step 3: Write the additive inverse using these new terms.
So, the additive inverse of [tex]\(9x^3 - 7x^2 - 13x + 2\)[/tex] is:
[tex]\[
-9x^3 + 7x^2 + 13x - 2
\][/tex]
Step 4: Compare with the given options.
The correct answer from the options provided should match this polynomial:
1) [tex]\(9x^2 + 7x^2 + 13x - 2\)[/tex] - Incorrect (wrong polynomial structure)
2) [tex]\(-9x^3 + 7x^2 + 13x - 2\)[/tex] - Correct
3. Option 3 is missing, but let's proceed.
4) [tex]\(9x^3 + 7x^2 - 13x - 2\)[/tex] - Incorrect (not the additive inverse)
Thus, the correct answer is:
[tex]\[
\boxed{-9x^3 + 7x^2 + 13x - 2}
\][/tex]
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