- Distribute the negative sign to the second polynomial: \(3x^2 - 8x - 2x^2 - 8\).
- Group the like terms together based on the power of \(x\): \((3x^2 - 2x^2) - 8x - 8\).
- Combine the coefficients to simplify the \(x^2\) terms: \(x^2 - 8x - 8\).
- The simplified expression is \(\boxed{x^2 - 8x - 8}\).
Explanation
- Analysis of the Expression
The problem asks to simplify the expression \((3x^2 - 8x) - (2x^2 + 8)\). This involves polynomial subtraction. To subtract one polynomial from another, we need to subtract each corresponding term of the second polynomial from the first. This is equivalent to distributing the negative sign to every term inside the parentheses of the second polynomial.
- Removing Parentheses and Distributing the Negative Sign
First, we remove the parentheses. When we remove the parentheses from the first part, we have \(3x^2 - 8x\). When we remove the parentheses from the second part, the negative sign is applied to both \(2x^2\) and \(8\), resulting in \(-2x^2 - 8\). The expression now becomes:
\[3x^2 - 8x - 2x^2 - 8\]
- Grouping Like Terms
Next, we identify and group like terms, which are terms that share the same power of the variable \(x\).
- Terms with \(x^2\): \(3x^2\) and \(-2x^2\)
- Terms with \(x\): \(-8x\)
- Constant terms: \(-8\)
Grouping these together, we have:
\[(3x^2 - 2x^2) - 8x - 8\]
- Simplification and Final Result
Now, we combine the coefficients of the grouped terms.
- For the \(x^2\) terms: \(3 - 2 = 1\), so \(3x^2 - 2x^2 = x^2\).
- The remaining terms \(-8x\) and \(-8\) cannot be simplified further as they are not like terms.
Putting it all together, the simplified expression is:
\[x^2 - 8x - 8\]
Thus, the final answer is \(\boxed{x^2 - 8x - 8}\).
Examples
Polynomial subtraction is useful in various real-world scenarios, such as managing budgets or tracking inventory. For instance, if a company's total revenue over a period is modeled by the function \(R(x) = 3x^2 - 8x\) and its expenses are modeled by \(E(x) = 2x^2 + 8\), where \(x\) represents units produced, the company's profit \(P(x)\) is calculated by subtracting expenses from revenue: \(P(x) = R(x) - E(x) = (3x^2 - 8x) - (2x^2 + 8)\). Simplifying this expression helps the manager quickly determine the net gain or loss function as \(P(x) = x^2 - 8x - 8\).