Factor The Expression $x^2 - 100$.$(x \square 10)(x \square 10)$

Feb 26, 2025 by ADMIN 69 views

**Factoring the Expression: A Step-by-Step Guide**

=====================================================

Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. In this article, we will focus on factoring the expression $x^2 - 100$, which can be written as $(x \square 10)(x \square 10)$. Factoring this expression will help us understand the underlying structure of the polynomial and make it easier to solve equations and inequalities involving this expression.

Understanding the Expression

The given expression is a quadratic expression in the form of $x^2 - 100$. This expression can be factored using the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$. In this case, we can rewrite the expression as $(x \square 10)(x \square 10)$, where $\square$ represents the operation of subtraction.

Factoring the Expression

To factor the expression, we need to find two binomials whose product is equal to the given expression. We can start by identifying the two terms in the expression, which are $x^2$ and $-100$. We can then look for two binomials whose product is equal to these two terms.

Using the difference of squares formula, we can rewrite the expression as $(x + 10)(x - 10)$. This is because $(x + 10)(x - 10) = x^2 - 100$, which is the given expression.

Verifying the Factored Form

To verify that the factored form is correct, we can multiply the two binomials together and check if we get the original expression. Multiplying $(x + 10)(x - 10)$, we get:

(x + 10)(x - 10) = x^2 - 10x + 10x - 100

Simplifying the expression, we get:

x^2 - 100

This shows that the factored form $(x + 10)(x - 10)$ is indeed correct.

Conclusion

In this article, we have factored the expression $x^2 - 100$ using the difference of squares formula. We have shown that the factored form is $(x + 10)(x - 10)$ and verified that this form is correct by multiplying the two binomials together. Factoring expressions is an important concept in algebra that helps us understand the underlying structure of polynomials and make it easier to solve equations and inequalities.

Examples and Applications

Factoring expressions has many practical applications in mathematics and other fields. Here are a few examples:

Solving Equations: Factoring expressions can help us solve equations by setting each factor equal to zero and solving for the variable.
Graphing Functions: Factoring expressions can help us graph functions by identifying the x-intercepts and other key features of the graph.
Simplifying Expressions: Factoring expressions can help us simplify complex expressions by breaking them down into simpler components.

Tips and Tricks

Here are a few tips and tricks for factoring expressions:

Look for Common Factors: Before factoring an expression, look for common factors that can be factored out.
Use the Difference of Squares Formula: The difference of squares formula is a powerful tool for factoring expressions of the form $a^2 - b^2$.
Check Your Work: Always check your work by multiplying the factored form together to make sure it equals the original expression.

Common Mistakes to Avoid

Here are a few common mistakes to avoid when factoring expressions:

Not Checking Your Work: Failing to check your work can lead to incorrect factored forms.
Not Using the Difference of Squares Formula: Failing to use the difference of squares formula can make it difficult to factor expressions of the form $a^2 - b^2$.
Not Looking for Common Factors: Failing to look for common factors can make it difficult to factor expressions.

Conclusion

In conclusion, factoring expressions is an important concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. By understanding the underlying structure of the polynomial and using the difference of squares formula, we can factor expressions and make it easier to solve equations and inequalities. With practice and patience, you can become proficient in factoring expressions and apply this skill to a wide range of mathematical problems.

=====================================

Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. In our previous article, we discussed how to factor the expression $x^2 - 100$ using the difference of squares formula. In this article, we will answer some frequently asked questions about factoring expressions.

Q&A

Q: What is factoring an expression?

A: Factoring an expression involves expressing a given polynomial as a product of simpler polynomials. This can help us understand the underlying structure of the polynomial and make it easier to solve equations and inequalities.

Q: How do I factor an expression?

A: To factor an expression, you need to identify the underlying structure of the polynomial and use the appropriate factoring technique. This can involve using the difference of squares formula, the sum of squares formula, or other factoring techniques.

Q: What is the difference of squares formula?

A: The difference of squares formula is a powerful tool for factoring expressions of the form $a^2 - b^2$. It states that $a^2 - b^2 = (a + b)(a - b)$.

Q: How do I use the difference of squares formula?

A: To use the difference of squares formula, you need to identify the two terms in the expression that are being subtracted. You can then rewrite the expression as $(a + b)(a - b)$, where $a$ and $b$ are the two terms.

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

A: Some common mistakes to avoid when factoring expressions include:

Not checking your work
Not using the difference of squares formula when necessary
Not looking for common factors
Not simplifying the expression after factoring

Q: How do I check my work when factoring expressions?

A: To check your work when factoring expressions, you need to multiply the factored form together and make sure it equals the original expression. This can help you identify any errors in your factoring.

Q: What are some real-world applications of factoring expressions?

A: Factoring expressions has many real-world applications, including:

Solving equations and inequalities
Graphing functions
Simplifying complex expressions
Identifying the underlying structure of polynomials

Q: Can you give me some examples of factoring expressions?

A: Here are a few examples of factoring expressions:

Factoring the expression $x^2 - 100$ using the difference of squares formula
Factoring the expression $x^2 + 10x + 25$ using the perfect square trinomial formula
Factoring the expression $x^2 - 4x + 4$ using the perfect square trinomial formula

Conclusion

Additional Resources

If you are looking for additional resources to help you learn about factoring expressions, here are a few suggestions:

Textbooks: There are many textbooks available that cover the topic of factoring expressions, including "Algebra" by Michael Artin and "College Algebra" by James Stewart.
Online Resources: There are many online resources available that cover the topic of factoring expressions, including Khan Academy, Mathway, and Wolfram Alpha.
Practice Problems: Practice problems are an essential part of learning about factoring expressions. You can find practice problems in textbooks, online resources, and practice problem books.