Importance of Simplifying Algebraic Expressions

Oct 13, 2025 by wbrum.org 107 views

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves combining like terms, removing parentheses, and rearranging the expression to make it easier to work with. In this article, we will focus on finding the difference between two algebraic expressions and simplifying the resulting expression.

The Problem

The problem we are given is to find the difference between two algebraic expressions:

$\left(t^4 - 1.5t^2 + t\right) - \left(12 - 9.5t^2 - 7t\right)$

Step 1: Distribute the Negative Sign

To find the difference between the two expressions, we need to distribute the negative sign to each term inside the second expression. This will change the sign of each term.

$\left(t^4 - 1.5t^2 + t\right) - \left(12 - 9.5t^2 - 7t\right) = t^4 - 1.5t^2 + t - 12 + 9.5t^2 + 7t$

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

$t^4 - 1.5t^2 + t - 12 + 9.5t^2 + 7t = t^4 + (9.5t^2 - 1.5t^2) + (t + 7t) - 12$

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by removing parentheses and combining the constants.

$t^4 + (9.5t^2 - 1.5t^2) + (t + 7t) - 12 = t^4 + 8t^2 + 8t - 12$

Conclusion

In conclusion, the difference between the two algebraic expressions is $t^4 + 8t^2 + 8t - 12$ . This expression is simplified by combining like terms and removing parentheses.

Final Answer

The final answer is: $\boxed{t^4 + 8t^2 + 8t - 12}$

Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential step in solving equations and inequalities. It helps to make the expression easier to work with and can often lead to a solution. In this article, we have seen how to simplify an algebraic expression by combining like terms and removing parentheses.

Tips for Simplifying Algebraic Expressions

Here are some tips for simplifying algebraic expressions:

Combine like terms by adding or subtracting the coefficients of the same variable raised to the same power.
Remove parentheses by distributing the negative sign to each term inside the parentheses.
Simplify the expression by combining the constants.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying algebraic expressions:

Failing to combine like terms.
Failing to remove parentheses.
Simplifying the expression incorrectly.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In economics, algebraic expressions are used to model the behavior of economic systems.

Conclusion

In conclusion, simplifying algebraic expressions is an essential step in solving equations and inequalities. It helps to make the expression easier to work with and can often lead to a solution. By following the tips and avoiding common mistakes, you can simplify algebraic expressions with ease.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics and has many real-world applications. By mastering this skill, you can solve equations and inequalities with ease and apply mathematical concepts to real-world problems.

References

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Introduction

In our previous article, we discussed how to simplify algebraic expressions by combining like terms and removing parentheses. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the difference between combining like terms and removing parentheses?

A: Combining like terms involves adding or subtracting the coefficients of the same variable raised to the same power. Removing parentheses involves distributing the negative sign to each term inside the parentheses.

Q: How do I know which terms to combine?

A: To combine like terms, you need to identify the terms that have the same variable raised to the same power. For example, in the expression $2x + 3x$ , the terms $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: Can I combine terms with different variables?

A: No, you cannot combine terms with different variables. For example, in the expression $2x + 3y$ , the terms $2x$ and $3y$ are not like terms because they have different variables.

Q: How do I remove parentheses?

A: To remove parentheses, you need to distribute the negative sign to each term inside the parentheses. For example, in the expression $(2x + 3y) - (4x - 2y)$ , you would distribute the negative sign to each term inside the second parentheses to get $2x + 3y - 4x + 2y$ .

Q: Can I simplify an expression with multiple parentheses?

A: Yes, you can simplify an expression with multiple parentheses by removing each set of parentheses one at a time. For example, in the expression $(2x + 3y) - (4x - 2y) + (x - 2y)$ , you would remove the first set of parentheses to get $2x + 3y - 4x + 2y$ , then remove the second set of parentheses to get $-2x + 5y$ , and finally remove the third set of parentheses to get $-2x + 5y$ .

Q: How do I know if an expression is simplified?

A: An expression is simplified when there are no like terms that can be combined. For example, in the expression $2x + 3x$ , the terms $2x$ and $3x$ are like terms, so the expression is not simplified. However, in the expression $2x + 4y$ , there are no like terms, so the expression is simplified.

Q: Can I simplify an expression with variables raised to different powers?

A: Yes, you can simplify an expression with variables raised to different powers by combining like terms. For example, in the expression $2x^2 + 3x^2$ , the terms $2x^2$ and $3x^2$ are like terms, so you can combine them to get $5x^2$ .

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to combine the fractions by finding a common denominator. For example, in the expression $\frac{1}{2} + \frac{1}{3}$ , you would find a common denominator of 6 to get $\frac{3}{6} + \frac{2}{6}$ , then combine the fractions to get $\frac{5}{6}$ .

Conclusion

In conclusion, simplifying algebraic expressions is an essential step in solving equations and inequalities. By following the tips and avoiding common mistakes, you can simplify algebraic expressions with ease. We hope this Q&A article has helped to clarify any questions you may have had about simplifying algebraic expressions.