- Identify the polynomials to be added: \(P(x) = 11x^2 - 5\) and \(Q(x) = x + 4\).
- Combine like terms by summing coefficients for the quadratic, linear, and constant parts: \(11x^2\), \(x\), and \(-5 + 4\).
- Simplify the resulting expression to get \(11x^2 + x - 1\).
- Identify the correct option: \(\boxed{c}\).
Explanation
- Understanding the Problem
We are given two polynomials: \(P(x) = 11x^2 - 5\) and \(Q(x) = x + 4\). The task is to find their sum, denoted by \(P(x) + Q(x)\). To perform this addition, we must group the terms with the same power of \(x\), which is a standard method for combining like terms in polynomials.
- Adding the Polynomials
To add the polynomials, we combine the coefficients of the corresponding terms:
- The quadratic term (\(x^2\)): The coefficient in \(P(x)\) is 11, and in \(Q(x)\) it is 0. Adding these, we get \(11x^2\).
- The linear term (\(x\)): The coefficient in \(P(x)\) is 0, and in \(Q(x)\) it is 1. Adding these, we get \(1x\), or simply \(x\).
- The constant term: The constant in \(P(x)\) is -5, and in \(Q(x)\) it is 4. Adding these, we get \(-5 + 4 = -1\).
- Conclusion and Final Answer Selection
Combining all the results from the previous step, we form the sum:
\(P(x) + Q(x) = 11x^2 + x - 1\).
Comparing this to the given choices:
A\(10x^2 - 9\)
B\(12x^2 - 1\)
C\(11x^2 + x - 1\)
D\(11x^2 - x - 9\)
The correct option is c.
Examples
In real-world applications, polynomial addition is used in physics and engineering to combine forces or model the total cost of production. For example, if one function represents the fixed cost for building a factory and another represents the variable cost of materials based on production volume, adding these two polynomials allows you to determine the total cost function for your business. Mastery of combining like terms is essential for accurate modeling and decision-making in such scenarios.