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Chapter 10: Problem 506

Add or subtract the polynomials. $$\left(4 a^{2}+9 a-11\right)+\left(6 a^{2}-5 a+10\right)$$

Short Answer

Expert verified
\(10a^2 + 4a - 1\)

Step by step solution

01

Identify the like terms

To add or subtract polynomials, identify the like terms. Like terms have the same variable raised to the same power. For the given polynomials, the like terms are: - Terms with \(a^2\): \(4a^2\) and \(6a^2\)- Terms with \(a\): \(9a\) and \(-5a\)- Constant terms: \(-11\) and \(10\)
02

Add the like terms

Now, add the coefficients of the like terms:- For \(a^2\): \(4a^2 + 6a^2 = 10a^2\)- For \(a\): \(9a - 5a = 4a\)- For the constants: \(-11 + 10 = -1\)
03

Combine the results

Combine the results from adding the like terms to get the final polynomial:- \(10a^2 + 4a - 1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
When adding or subtracting polynomials, understanding 'like terms' is crucial. Like terms are terms that have the same variable raised to the same power. For example, in the polynomial expression from the exercise, the terms with the variable and power combinations like \(a^2\) must be added together. Similarly, terms with just \(a\) and the constant terms should be combined. Recognizing like terms helps in simplifying the expression easily. Identifying like terms ensures that you do not mix terms inappropriately and helps in maintaining the correct structure of the polynomial.
Coefficients
In the context of polynomials, coefficients refer to the numerical part of each term. For instance, in the term \(4a^2\), the number 4 is the coefficient. Knowing how to work with coefficients is essential because when you add or subtract polynomials, you are primarily dealing with the coefficients of like terms.
In the exercise, consider the coefficients of \(4a^2 \ and \ 6a^2\). Adding these coefficients (4 and 6) gives us 10, resulting in the term \(10a^2\). This process applies to all other like terms in the polynomial. Always ensure you're adding or subtracting the coefficients of like terms consistently.
Combining Polynomials
Combining polynomials involves adding or subtracting terms systematically. Start by grouping the like terms together. For example, in the given exercise, terms \(4a^2\ and \ 6a^2\) are grouped together because they are like terms.
Once you've grouped all the like terms, add or subtract their coefficients accordingly. Finally, combine the results from each group to form a new polynomial. Make sure you follow the correct order and do not mix terms. Combining the results correctly gives a simplified polynomial, as shown in the exercise, resulting in the final expression \(10a^2 + 4a - 1\). By following these steps, working with polynomials becomes straightforward and manageable.

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Most popular questions from this chapter

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