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Chapter 5: Problem 87

Add. $$ \left(x^{2}+2 x-3 x y-7\right)+\left(-3 x^{2}-x+2 y^{2}+6\right) $$

Short Answer

Expert verified
Combine like terms: \textbf{Simplified Expression:} \text{ \text{Simplified Expression:} -2x^2 + x - 3xy + 2y^2 - 1}

Step by step solution

01

Identify like terms

List out the like terms from each polynomial. Like terms are terms with the same variables raised to the same power. In this case, identify the coefficients of each variable term.
02

Combine like terms

Add the coefficients of the like terms together. For example, combine the coefficients of each term involving the same power of x or y.
03

Simplify the expression

Write the resulting simplified polynomial, combining all like terms into single terms. Make sure to keep track of the positive and negative signs while combining.

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Key Concepts

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

like terms
Before you can combine terms in a polynomial, you need to **identify like terms**. Like terms have the same variables raised to the same powers. For instance, in the given problem, the terms involving the variable \(x^2\) are \(x^2\) and \(-3x^2\). These are considered like terms because they both have the same variable \(x\) raised to the same power \(2\). The constants or numbers without variables, like \(-7\) and \(6\), are also like terms.
combining coefficients
Once you identify the like terms in the polynomial, the next step is to combine their **coefficients**. Coefficients are the numerical part of the terms. For example, in the term \(2x\), the coefficient is \(2\). In the term \(-3x^2\), the coefficient is \(-3\). To combine coefficients, you add or subtract them. In our example, combining \(x^2\) and \(-3x^2\) would give you \[(1-3)x^2=-2x^2\]. Similarly, you would add the constants \(-7\) and \(6\) as \[-7+6=-1\].
simplifying expressions
After combining all like terms, the next step is to **simplify the expression**. This involves ensuring all like terms have been combined into single terms. For example, given \[x^2+2x-3xy-7 + (-3x^2-x+2y^2+6)\], combining the like terms, we get \[-2x^2 +x-3xy+2y^2-1\]. At this stage, make sure to handle both positive and negative signs correctly, ensuring no terms are left uncombined.
variables and powers
Finally, understanding how **variables and powers** work is crucial. Variables represent unknown values and are often denoted as letters like \(x\) or \(y\). When they are raised to a power, such as \(x^2\) or \(y^2\), it means the variable is multiplied by itself that many times. In our example, \(2y^2\) means \(2\times y\times y\). Recognizing and correctly handling these variables and their powers will help you accurately combine and simplify the polynomials.

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