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Chapter 2: Problem 93

Simplify by combining like terms whenever possible. $$-5 s^{2}+3 s t-s^{2}+6 s^{2}$$

Short Answer

Expert verified
The simplified expression is 3st.

Step by step solution

01

Identify Like Terms

Look at the given expression ewline ewline -5 s^{2} + 3 s t - s^{2} + 6 s^{2}. Identify the terms involving the same variables raised to the same power. In this case, notice there are three terms involving s^{2}: ewlineewline -5 s^{2}, - s^{2}, and 6 s^{2}. Recognize that 3 s t is a different term.
02

Combine Like Terms

Combine the coefficients of the like terms. Add the coefficients of s^{2}: -5 + -1 + 6. This simplifies to 0. The term with 3 s t remains unchanged, as there are no other similar terms.
03

Simplify Expression

Write down the simplified expression combining the calculated coefficients. ewline -5 s^{2} + 3 s t - s^{2} + 6 s^{2} ewline = 0 s^{2} + 3 s t ewline = 3 s t. Hence, the combined and simplified expression is ewline 3 s t.

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Key Concepts

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

Combining Like Terms
The first step in simplifying algebraic expressions is to combine like terms. Like terms are those terms that have the same variables raised to the same power. For instance, in the expression ewline -5s^2 + 3st - s^2 + 6s^2ewline, the terms ewline -5s^2ewline, ewline -s^2ewline, and ewline 6s^2ewline are like terms because each one involves ewline s^2ewline. However, ewline 3stewline is not a like term with the others because it involves different variables and powers (s and t). By identifying and combining like terms, we can simplify the expression step-by-step, which makes calculations easier and clearer.
Mathematical Coefficients
Coefficients are numerical factors in front of variables in algebraic terms. In the terms ewline -5s^2ewline, ewline -s^2ewline, and ewline 6s^2ewline, the coefficients are ewline -5ewline, ewline -1ewline, and ewline 6ewline, respectively. When combining like terms, it鈥檚 important to add or subtract these coefficients while keeping the variable part the same. So for ewline -5s^2 + (-1s^2) + 6s^2ewline, we combine the coefficients: ewline -5 + (-1) + 6 = 0ewline. Understanding how to manipulate coefficients is crucial in simplifying algebraic expressions.
Polynomial Simplification
Simplifying polynomials involves several steps: identifying like terms, combining coefficients, and rewriting the expression in its simplest form. In our example, we started with the expression ewline -5s^2 + 3st - s^2 + 6s^2ewline. We identified the like terms, combined the coefficients of ewline s^2ewline to get ewline 0s^2 ewline, and kept the remaining term ewline 3stewline unchanged because it had no like terms. Thus, the simplified polynomial is ewline 3stewline. The objective is to make the expression as concise as possible without changing its value. This technique is widely used in algebra to make expressions easier to work with in equations and problem-solving.

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