For All $x$, What Is $(x^5 + 6x^3 + X - 7) - (x^5 - X^4 + X^3 + 6$\]?

Feb 28, 2025 by ADMIN 70 views

Simplifying Algebraic Expressions: A Step-by-Step Guide

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given expression $(x^5 + 6x^3 + x - 7) - (x^5 - x^4 + x^3 + 6)$ . We will break down the expression into manageable parts, apply the rules of algebra, and arrive at the simplified form.

Understanding the Expression

The given expression is a combination of two polynomials, each with its own set of terms. To simplify the expression, we need to first understand the rules of combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two sets of like terms: $x^5$ , $x^3$ , and $x$ .

Distributing the Negative Sign

The first step in simplifying the expression is to distribute the negative sign to each term inside the second set of parentheses. This will change the sign of each term, effectively multiplying each term by $-1$ . The expression becomes:

$(x^5 + 6x^3 + x - 7) - (x^5 - x^4 + x^3 + 6) = (x^5 + 6x^3 + x - 7) + (-x^5 + x^4 - x^3 - 6)$

Combining Like Terms

Now that we have distributed the negative sign, we can combine like terms. We will group the terms with the same variable raised to the same power. The expression becomes:

$(x^5 + 6x^3 + x - 7) + (-x^5 + x^4 - x^3 - 6) = (x^5 - x^5) + (6x^3 - x^3) + (x^4) + (x - x) + (-7 - 6)$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression further. We will eliminate any terms that cancel each other out. The expression becomes:

$(x^5 - x^5) + (6x^3 - x^3) + (x^4) + (x - x) + (-7 - 6) = 0 + 5x^3 + x^4 + 0 - 13$

Final Simplified Form

The final simplified form of the expression is:

$5x^3 + x^4 - 13$

Conclusion

In this article, we have explored the process of simplifying algebraic expressions, focusing on the given expression $(x^5 + 6x^3 + x - 7) - (x^5 - x^4 + x^3 + 6)$ . We have broken down the expression into manageable parts, applied the rules of algebra, and arrived at the simplified form. The final simplified form of the expression is $5x^3 + x^4 - 13$ . This demonstrates the importance of simplifying algebraic expressions, as it allows us to better understand and work with mathematical concepts.

Tips and Tricks

When simplifying algebraic expressions, always start by distributing the negative sign to each term inside the second set of parentheses.
Group like terms together to make it easier to combine them.
Eliminate any terms that cancel each other out.
Double-check your work to ensure that the final simplified form is correct.

Common Mistakes

Failing to distribute the negative sign to each term inside the second set of parentheses.
Not grouping like terms together.
Not eliminating terms that cancel each other out.
Not double-checking the final simplified form.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems. In economics, algebraic expressions are used to model and analyze economic systems.

Final Thoughts

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions. Remember to always distribute the negative sign, group like terms together, eliminate terms that cancel each other out, and double-check your work. With practice and patience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most challenging math problems.

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given expression $(x^5 + 6x^3 + x - 7) - (x^5 - x^4 + x^3 + 6)$ . We broke down the expression into manageable parts, applied the rules of algebra, and arrived at the simplified form. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to distribute the negative sign to each term inside the second set of parentheses.

Q: How do I know which terms to combine?

A: To combine like terms, you need to group the terms with the same variable raised to the same power. For example, in the expression $x^5 + 6x^3 + x - 7 - x^5 + x^4 - x^3 - 6$ , you can group the terms as follows:

$x^5$ and $-x^5$ (these terms cancel each other out)
$6x^3$ and $-x^3$ (these terms combine to form $5x^3$ )
$x^4$ (this term remains as is)
$x$ and $-x$ (these terms cancel each other out)
$-7$ and $-6$ (these terms combine to form $-13$ )

Q: What is the difference between combining like terms and eliminating terms that cancel each other out?

A: Combining like terms involves adding or subtracting terms with the same variable raised to the same power. Eliminating terms that cancel each other out involves removing terms that have the same variable raised to the same power, but with opposite coefficients.

Q: How do I know if I have simplified an expression correctly?

A: To ensure that you have simplified an expression correctly, you need to double-check your work. This involves re-reading the expression and verifying that you have combined like terms correctly and eliminated terms that cancel each other out.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

Failing to distribute the negative sign to each term inside the second set of parentheses
Not grouping like terms together
Not eliminating terms that cancel each other out
Not double-checking the final simplified form

Q: How do I apply the rules of algebra to simplify an expression?

A: To apply the rules of algebra to simplify an expression, you need to follow these steps:

Distribute the negative sign to each term inside the second set of parentheses
Group like terms together
Combine like terms
Eliminate terms that cancel each other out
Double-check the final simplified form

Real-World Applications

Tips and Tricks

When simplifying algebraic expressions, always start by distributing the negative sign to each term inside the second set of parentheses.
Group like terms together to make it easier to combine them.
Eliminate any terms that cancel each other out.
Double-check your work to ensure that the final simplified form is correct.

Common Mistakes

Failing to distribute the negative sign to each term inside the second set of parentheses.
Not grouping like terms together.
Not eliminating terms that cancel each other out.
Not double-checking the final simplified form.