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Chapter 3: Problem 76

Find the sum or difference. \(\left(x^2+2\right)+\left(2 x^2-x\right)\)

Short Answer

Expert verified
The simplified form of the given sum is \(3x^2 - x + 2\).

Step by step solution

01

Identify Like Terms

In each polynomial, we look for terms that have the same variable and exponent. Here \(x^2\) in \(\left(x^2+2\right)\) and \(2 x^2\) in \(\left(2 x^2-x\right)\) are like terms.
02

Combine Like Terms

We now add the like terms. On adding, we get \(x^2 + 2 x^2\) which equals to \(3x^2\). Now, we include the remaining terms in the polynomials, adding \(-x\) and \(2\) from each of the two given polynomials to get the sum.
03

Write the Final Answer

On putting it all together, we get \(3x^2 - x + 2\). This is our simplified polynomial.

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential skill when working with algebraic expressions. It involves identifying and adding or subtracting terms that have identical variables raised to the same power. These are called 'like terms'. For instance, in the polynomial addition problem
\[\begin{equation}\left(x^2+2\right)+\left(2x^2-x\right)\end{equation}\],
we identify \(x^2\) in the first polynomial and \(2x^2\) in the second as like terms because both contain the variable \(x\) raised to the second power.
Simplifying Expressions
Simplification involves reducing an algebraic expression to its simplest form. This means combining like terms, as shown in the exercise above, and performing any applicable arithmetic operations. After combining like terms (\(x^2 + 2x^2\)), the expression will often become easier to work with or solve. Simplification does not change the value of the expression; it merely presents it in a more digestible form. In the exercise, after combining like terms, we are left with the much simpler expression \[\begin{equation}3x^2 - x + 2\end{equation}\], making it clear what the polynomial's terms are.
Algebraic Operations
Algebraic operations are the procedures we use to manipulate algebraic expressions, including addition, subtraction, multiplication, and division. In the context of polynomial addition, we are primarily concerned with addition and subtraction. To properly perform these operations, we follow the order of operations and combine like terms. The exercise provided demonstrates addition where the like terms, both including the variable \(x\) raised to the same power, are added together. Understanding how to correctly carry out algebraic operations is crucial for simplifying expressions and solving equations.

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Most popular questions from this chapter

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