- Remove parentheses to express the sum as \(8x^2 - 9y^2 - 4x + x^2 - 3y^2 - 7x\).
- Group like terms: \((8x^2 + x^2) + (-9y^2 - 3y^2) + (-4x - 7x)\).
- Add the coefficients to simplify: \(9x^2 - 12y^2 - 11x\).
- The correct option is a. \(\boxed{9x^2 - 12y^2 - 11x}\).
Explanation
- Problem Analysis
To find the sum of the polynomials \((8x^2 - 9y^2 - 4x)\) and \((x^2 - 3y^2 - 7x)\), we need to group and combine the terms that have the same variables and powers. These are known as 'like terms'.
The given expression is:
\[(8x^2 - 9y^2 - 4x) + (x^2 - 3y^2 - 7x)\]
- Combining Like Terms
First, we remove the parentheses, since adding the second polynomial does not change the signs of its terms:
\[8x^2 - 9y^2 - 4x + x^2 - 3y^2 - 7x\]
Next, we group the like terms together based on their variables and exponents:
- Terms with \(x^2\): \(8x^2 + x^2\)
- Terms with \(y^2\): \(-9y^2 - 3y^2\)
- Terms with \(x\): \(-4x - 7x\)
Now we combine these groups:
- For \(x^2\): \((8 + 1)x^2 = 9x^2\)
- For \(y^2\): \((-9 - 3)y^2 = -12y^2\)
- For \(x\): \((-4 - 7)x = -11x\)
- Conclusion
By summing the coefficients of the grouped terms, we obtain:
\[9x^2 - 12y^2 - 11x\]
Comparing this result to the given options:
A\(9x^2 - 12y^2 - 11x\)
B\(9x^2 - 6y^2 + 3x\)
C\(9x^2 - 12y^2 + 3x\)
D\(7x^2 - 6y^2 + 3x\)
The result matches option (a). Thus, the sum of the polynomials is \(\boxed{9x^2 - 12y^2 - 11x}\).
Examples
Algebraic expressions like this are used in physics and engineering to represent combined forces or physical quantities. For instance, if one force is represented by \(8x^2 - 9y^2 - 4x\) and another by \(x^2 - 3y^2 - 7x\), finding the total resultant force involves summing these polynomials just as we did here. Understanding how to group and combine terms allows us to simplify complex systems and find total values efficiently in many practical fields.