- Polynomial subtraction requires distributing the negative sign to all terms of the second polynomial, effectively adding its additive inverse.
- The additive inverse of \((7n^2 - 5n + 6)\) is \((-7n^2 + 5n - 6)\).
- Anna used \((-7n^2 + 5n + 6)\) in her addition, meaning she failed to change the sign of the constant term.
- This confirms that she made a mistake when finding the additive inverse of the second polynomial, which corresponds to option \(\boxed{c}\).
Explanation
- Analysis of the Polynomial Subtraction Error
The problem asks to identify the error Anna made when subtracting the polynomial \((7n^2 - 5n + 6)\) from \((5n^2 + 4n - 9)\).
To perform this subtraction correctly, we should write it as:
\[(5n^2 + 4n - 9) - (7n^2 - 5n + 6)\]
Subtracting a polynomial is equivalent to adding its additive inverse. The additive inverse of a polynomial is obtained by changing the sign of every term within it.
The additive inverse of \((7n^2 - 5n + 6)\) is:
\[-(7n^2 - 5n + 6) = -7n^2 + 5n - 6\]
Therefore, the correct subtraction is:
\[(5n^2 + 4n - 9) + (-7n^2 + 5n - 6)\]
Now, let's examine Anna's work:
\[ \begin{array}{r} 5 n^2 + 4 n + (-9) \ +\left(-7 n^2\right) + 5 n + 6 \\ -2 n^2 + 9 n + (-3) \end{array} \]
Anna's second line is \(-7n^2 + 5n + 6\). Comparing this to the correct additive inverse \(-7n^2 + 5n - 6\), we can see that Anna kept the constant term of the second polynomial as \(+6\) instead of changing it to \(-6\). This indicates she made a mistake in finding the additive inverse of the second polynomial.
- Identifying the Error and Conclusion
Comparing the options provided with our findings:
AShe did not align like terms correctly: Anna aligned \(n^2\) with \(n^2\), \(n\) with \(n\), and constants with constants. This is not the error.
BShe forgot to use the additive inverse of \((5 n^2 + 4 n - 9)\): Anna correctly used the first polynomial as it is. This is not the error.
CShe made a mistake when finding the additive inverse of \((7 n^2 - 5 n + 6)\): This matches our finding that she incorrectly used \(+6\) instead of \(-6\) for the constant term.
Thus, Anna's error was in calculating the additive inverse of \((7n^2 - 5n + 6)\). Specifically, she failed to change the sign of the constant term \(6\) to \(-6\). Therefore, the correct option is c.
Examples
Understanding how to correctly subtract polynomials is essential in algebra, as it appears frequently when simplifying complex algebraic expressions, solving equations, or analyzing functions. For instance, in economics, if you have a revenue function \(R(n) = 5n^2 + 4n - 9\) and a cost function \(C(n) = 7n^2 - 5n + 6\), the profit function is the difference between revenue and cost: \(P(n) = R(n) - C(n)\). Correctly distributing the negative sign across every term of the cost function is crucial for calculating the accurate profit. If you miss a sign, as Anna did, you will end up with an incorrect result, which would lead to faulty financial planning or data analysis.