- Identify like terms in the polynomials: \(x^2\) terms, \(x\) terms, and constant terms.
- Add coefficients of the \(x^2\) terms: \(8 - 6 = 2\).
- Add coefficients of the \(x\) terms: \(-8 + 3 = -5\).
- Combine the results with the constant term: \(\boxed{2x^2 - 5x - 8}\).
Explanation
- Identifying Like Terms
To find the sum of the polynomials \((8x^2 - 8x - 8)\) and \((-6x^2 + 3x)\), we need to identify and combine like terms. Like terms are terms that have the same variables raised to the same exponents. In these polynomials, we have:
- The \(x^2\) terms: \(8x^2\) and \(-6x^2\).
- The \(x\) terms: \(-8x\) and \(3x\).
- The constant terms: \(-8\) and \(0\) (as the second polynomial does not have a constant term).
- Combining Coefficients
Now, we combine the like terms by adding their coefficients:
- For the \(x^2\) terms: \(8x^2 + (-6x^2) = (8 - 6)x^2 = 2x^2\).
- For the \(x\) terms: \(-8x + 3x = (-8 + 3)x = -5x\).
- For the constants: \(-8 + 0 = -8\).
- Final Summation
Finally, we assemble the result by placing these combined terms into a single expression:
\(2x^2 - 5x - 8\). This is the simplified sum of the given polynomials. Thus, \((8x^2 - 8x - 8) + (-6x^2 + 3x) = \boxed{2x^2 - 5x - 8}\).
Examples
In engineering, when designing systems like electrical circuits, you might represent the total impedance or voltage across different components as polynomials. If one component contributes \(8x^2 - 8x - 8\) units of voltage and another contributes \(-6x^2 + 3x\) units, finding the net voltage requires adding these polynomials. By combining like terms, you quickly find the total system response, which is crucial for efficient design and analysis of complex systems.