Eliminate unnecessary components

Oct 13, 2025 by magicalceylontours.com 13 views

Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves combining like terms and eliminating any unnecessary components to make the expression more manageable and easier to work with. In this article, we will focus on simplifying the given expression $\left(3x^2 - 2x + 2\right) - \left(x^2 + 5x - 5\right)$ using the rules of algebra.

Understanding the Expression

The given expression is a difference of two polynomials, which means we need to subtract the second polynomial from the first one. To simplify this expression, we will first expand the parentheses and then combine like terms.

Expanding the Parentheses

To expand the parentheses, we need to distribute the negative sign to each term inside the second set of parentheses. This will give us:

$\left(3x^2 - 2x + 2\right) - \left(x^2 + 5x - 5\right)$

$= 3x^2 - 2x + 2 - x^2 - 5x + 5$

Combining Like Terms

Now that we have expanded the parentheses, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with $x^2$ , two terms with $x$ , and two constant terms.

$= 3x^2 - x^2 - 2x - 5x + 2 + 5$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by combining the coefficients of like terms. The coefficients are the numbers that multiply the variables.

$= 2x^2 - 7x + 7$

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it involves combining like terms and eliminating any unnecessary components. In this article, we have simplified the given expression $\left(3x^2 - 2x + 2\right) - \left(x^2 + 5x - 5\right)$ using the rules of algebra. We expanded the parentheses, combined like terms, and simplified the expression to get the final result of $2x^2 - 7x + 7$ .

Tips for Simplifying Algebraic Expressions

Here are some tips for simplifying algebraic expressions:

Combine like terms: Like terms are terms that have the same variable raised to the same power. Combine the coefficients of like terms to simplify the expression.
Eliminate unnecessary components: Eliminate any unnecessary components, such as zero terms or terms that cancel each other out.
Use the distributive property: Use the distributive property to expand the parentheses and simplify the expression.
Check your work: Check your work by plugging in values or using a calculator to verify the result.

Examples of Simplifying Algebraic Expressions

Here are some examples of simplifying algebraic expressions:

Example 1: Simplify the expression $\left(2x^2 + 3x - 1\right) - \left(x^2 + 2x + 2\right)$ .
Example 2: Simplify the expression $\left(x^2 - 2x + 1\right) + \left(2x^2 + 3x - 2\right)$ .
Example 3: Simplify the expression $\left(3x^2 - 2x + 2\right) - \left(x^2 + 5x - 5\right)$ .

Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions has many applications in mathematics and other fields. Here are some examples:

Solving equations: Simplifying algebraic expressions is essential for solving equations and inequalities.
Graphing functions: Simplifying algebraic expressions is necessary for graphing functions and understanding their behavior.
Optimization: Simplifying algebraic expressions is crucial for optimization problems, where we need to find the maximum or minimum value of a function.
Computer science: Simplifying algebraic expressions is used in computer science to optimize algorithms and improve performance.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it involves combining like terms and eliminating any unnecessary components. In this article, we have simplified the given expression $\left(3x^2 - 2x + 2\right) - \left(x^2 + 5x - 5\right)$ using the rules of algebra. We expanded the parentheses, combined like terms, and simplified the expression to get the final result of $2x^2 - 7x + 7$ . Simplifying algebraic expressions has many applications in mathematics and other fields, and it is an essential skill for anyone working with algebraic expressions.