- Group the polynomials into like terms: \(x^2\), \(y^2\), and \(x\) terms.
- Sum the coefficients: \(8+1=9\), \(-9-3=-12\), and \(-4-7=-11\).
- Combine the coefficients back with their variables: \(9x^2 - 12y^2 - 11x\).
- Select the matching option, which is \(\boxed{9x^2 - 12y^2 - 11x}\).
Explanation
- Understanding the Problem and Identifying Like Terms
We are asked to find the sum of two polynomials: \(P_1 = 8x^2 - 9y^2 - 4x\) and \(P_2 = x^2 - 3y^2 - 7x\). The sum is given by \((8x^2 - 9y^2 - 4x) + (x^2 - 3y^2 - 7x)\). To add these polynomials, we need to group the "like terms," which are the terms that have the same variables and the same exponents.
- Combining Like Terms
We identify the like terms as follows:
- The \(x^2\) terms: \(8x^2\) and \(x^2\).
- The \(y^2\) terms: \(-9y^2\) and \(-3y^2\).
- The \(x\) terms: \(-4x\) and \(-7x\).
Now, we add the coefficients for each set of like terms:
- For \(x^2\): \(8 + 1 = 9\), so the term is \(9x^2\).
- For \(y^2\): \(-9 + (-3) = -12\), so the term is \(-12y^2\).
- For \(x\): \(-4 + (-7) = -11\), so the term is \(-11x\).
- Conclusion and Matching with Options
By combining these results, the sum of the polynomials is \(9x^2 - 12y^2 - 11x\). Comparing this result to the given options, we find that it matches the fourth option. Therefore, the final answer is \(9x^2 - 12y^2 - 11x\).
Examples
Adding polynomials is a fundamental operation in algebra used in various fields like physics and engineering. For example, if you are modeling the trajectory of an object moving in two dimensions, you might have one polynomial expression for the horizontal position and another for the vertical position, and you need to combine these expressions to simplify your analysis of the total path taken by the object. This technique of combining like terms ensures that you can simplify complex equations into more manageable forms.