- Distribute the negative sign to the second polynomial: \(-(5x^2 - 14y^2 - 9x) = -5x^2 + 14y^2 + 9x\).
- Group like terms: \((12x^2 - 5x^2) + (-11y^2 + 14y^2) + (-13x + 9x)\).
- Perform the subtraction and addition for each coefficient: \(7x^2 + 3y^2 - 4x\).
- The final answer is \(\boxed{7x^2 + 3y^2 - 4x}\).
Explanation
- Simplifying the Expression
To find the difference between the two polynomials \((12x^2 - 11y^2 - 13x)\) and \((5x^2 - 14y^2 - 9x)\), we need to subtract the second polynomial from the first. This is expressed as:
\[(12x^2 - 11y^2 - 13x) - (5x^2 - 14y^2 - 9x)\]
The first step is to distribute the negative sign across the second polynomial. This changes the sign of each term within the parentheses:
\[12x^2 - 11y^2 - 13x - 5x^2 + 14y^2 + 9x\]
Now, we group the like terms (terms with the same variable and exponent) together:
\[(12x^2 - 5x^2) + (-11y^2 + 14y^2) + (-13x + 9x)\]
Finally, we perform the arithmetic for each group of like terms:
- For the \(x^2\) terms: \(12 - 5 = 7\), so we have \(7x^2\).
- For the \(y^2\) terms: \(-11 + 14 = 3\), so we have \(3y^2\).
- For the \(x\) terms: \(-13 + 9 = -4\), so we have \(-4x\).
Combining these results, the simplified expression is \(7x^2 + 3y^2 - 4x\).
Examples
In everyday life, this process of combining like terms is used in budgeting and inventory management. For example, if you are calculating your total expenses after a refund, you might subtract the refunded items from your initial list of costs. Just like combining \(x^2\) terms with other \(x^2\) terms, you add or subtract the same types of costs (e.g., groceries, fuel, or rent) to find the net total, ensuring accuracy in your financial records.