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Chapter 4: Problem 79

Simplify the expression. $$ x^{2}-2 x-x^{2}+3 x+2 $$

Short Answer

Expert verified
The simplified form of the expression is \(x + 2\).

Step by step solution

01

Identify like terms

The given expression is \(x^{2}-2x-x^{2}+3x+2\). Here, the like terms are \(x^{2}\) and \(-x^{2}\), and \(-2x\) and \(3x\).
02

Combine like terms

Next, you combine like terms by adding or subtracting them. Adding \(x^{2}\) and \(-x^{2}\) results in 0. Also, \(-2x\) and \(3x\) combine to \(x\). The expression now becomes \(0 + x + 2\)
03

Simplify the expression

We can simplify \(0 + x + 2\) to \(x + 2\). So, the simplified form of the given expression is \(x + 2\).

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Key Concepts

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

Combining Like Terms
When you simplify an algebraic expression, one of the first tasks is to identify and combine like terms. But what are like terms? Simply put, like terms are terms in an expression that have the same variables raised to the same power. For example, in the expression \( x^2 - 2x - x^2 + 3x + 2 \), we see two sets of like terms: \( x^2 \) and \( -x^2 \), as well as \( -2x \) and \( 3x \).
Combining like terms involves adding or subtracting the numerical coefficients of these terms while keeping the common variable and exponent. This process helps to consolidate the expression into fewer terms.
  • For \( x^2 - x^2 \), the coefficients are 1 and -1, which add up to 0.
  • For \( -2x + 3x \), the coefficients are -2 and 3, which sum to 1, leaving us with just \( x \).
By combining like terms, the expression becomes easier to manage and understand.
Polynomial Simplification
Once like terms are combined, simplifying a polynomial involves rearranging and reducing the expression to its simplest form. In our example, the expression \( 0 + x + 2 \) simplifies quite nicely.
Initially, we might have had more terms, but by combining like terms, we reduced it significantly. Simplification requires continual scrutiny to see if we can make it clearer and shorter without changing the value.
  • First, combine like terms (covered in the previous section).
  • Then, eliminate zero terms. For instance, \( 0 + x + 2 \) simplifies to \( x + 2 \) because adding zero does not change the value.
Polynomial simplification seeks efficiency and clarity. The goal is having an expression where each term is necessary, clear, and simple.
Algebraic Operations
Within the world of algebra, operations involve addition, subtraction, multiplication, and division of algebraic expressions. These operations follow the basic rules of arithmetic, but they work with variables and constants. In simplifying \( x^2 - 2x - x^2 + 3x + 2 \), we engaged primarily in algebraic addition and subtraction.
  • **Addition:** This involves increasing a term by a specified quantity, such as combining \( -2x \) with \( 3x \) to get \( x \).
  • **Subtraction:** Subtraction removes an amount from a term, as we saw when subtracting \( x^2 \) from \( x^2 \) to cancel it out.
Understanding algebraic operations is crucial for simplifying expressions. It contributes to one's deeper knowledge of how expressions behave and transform through various operations. This comprehension forms the core of algebra, helping students solve more complex equations efficiently.

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