Problem 56 \(\left(2 r^{2}-3 r s-2 s^{2}\ri... [FREE SOLUTION]

Chapter 5: Problem 56

\(\left(2 r^{2}-3 r s-2 s^{2}\right)+\left(5 r^{2}-3 r s\right)\)

Short Answer

Expert verified

The simplified form of the expression is

Step by step solution

Identify like terms

Separate the polynomial into like terms. Like terms have the same combination of variables and exponents. For the given expression, group the terms as follows:

Combine like terms

Add the coefficients of the like terms together: i.e.:

Simplify the expression

After combining like terms, your simplified expression will be obtained. Simplify your result if possible to reach the final expression.

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Key Concepts

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

like terms

In polynomial addition, identifying like terms is an essential first step. Like terms are terms within an expression that have the same variables raised to the same powers.
For instance, in the polynomial expression \((2r^2 - 3rs - 2s^2) + (5r^2 - 3rs)\), we identify like terms:
- Terms with \({r^2}\) are \({2r^2}\) and \({5r^2}\)\(\(r^2\)\).
- Terms with \({rs}\) are \({-3rs}\) and \({-3rs}\)\(\(rs\)\).
- Term with \({s^2}\) is \({-2s^2}\).

By grouping terms with the same combination of variables and exponents, we simplify the process of addition in polynomials.

coefficients

A crucial part of polynomial addition is working with coefficients. Coefficients are the numerical parts of the terms. They multiply the variable part of the term.
For example, in the term \(({2 r^2})\), \({2}\) is the coefficient.

To add polynomials, we add the coefficients of like terms:
- For the term \({r^2}\), we perform \({2 + 5}\) to get \({7r^2}\).
- For the term \({rs}\), we perform \({-3 + -3}\) to get \({-6rs}\).
- For the term \({s^2}\), since there is only one term \({-2s^2}\), it remains unchanged.

Understanding and manipulating coefficients will allow you to successfully combine like terms.

simplifying expressions

Simplifying expressions involves combining like terms to create a more concise and easily understandable polynomial.
After identifying and adding like terms, we rewrite the polynomial with the summed coefficients.

For the given problem, after adding the coefficients of the like terms, we get:
- For \({r^2}\): \({2r^2 + 5r^2 = 7r^2}\)
- For \({rs}\): \({-3rs + -3rs = -6rs}\)
- For \({s^2}\): It stays as \({-2s^2}\)

Therefore, the simplified expression is \({7r^2 - 6rs - 2s^2}\).
Simplifying expressions makes the polynomial easier to work with and understand. It's an essential skill for higher-level math problems.

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\(\left(2 r^{2}-3 r s-2 s^{2}\right)+\left(5 r^{2}-3 r s\right)\)

Short Answer

Step by step solution

Identify like terms

Combine like terms

Simplify the expression

Key Concepts

like terms

coefficients

simplifying expressions

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