Your Answer Should Be A Polynomial In Standard Form. { (x-4)(x-6) = $}$ { \square$}$

Mar 13, 2025 by ADMIN 85 views

Expanding and Simplifying Algebraic Expressions: A Step-by-Step Guide

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and understanding how to expand and simplify them is crucial for solving equations and inequalities. In this article, we will focus on expanding and simplifying polynomial expressions, specifically the product of two binomials. We will use the given expression $(x-4)(x-6)$ as an example and guide you through the step-by-step process of expanding and simplifying it.

Understanding the Problem

The given expression is a product of two binomials, $(x-4)$ and $(x-6)$ . To expand and simplify this expression, we need to use the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b+c) = ab + ac$ . We will apply this property to each term in the first binomial and then multiply it by each term in the second binomial.

Expanding the Expression

To expand the expression, we will start by multiplying each term in the first binomial, $(x-4)$ , by each term in the second binomial, $(x-6)$ . We will use the distributive property to multiply each term:

(x-4)(x-6) = x(x-6) - 4(x-6)

Applying the Distributive Property

Now, we will apply the distributive property to each term in the expression:

x(x-6) = x^2 - 6x
- 4(x-6) = -4x + 24

Combining Like Terms

We will now combine like terms in the expression:

x^2 - 6x - 4x + 24

Simplifying the Expression

We will now simplify the expression by combining like terms:

x^2 - 10x + 24

Conclusion

In this article, we have expanded and simplified the given expression $(x-4)(x-6)$ using the distributive property and combining like terms. We have shown that the expanded and simplified form of the expression is $x^2 - 10x + 24$ . This process is essential for solving equations and inequalities, and it is a fundamental concept in algebra.

Tips and Tricks

When expanding and simplifying algebraic expressions, it is essential to use the distributive property and combine like terms.
Make sure to multiply each term in the first binomial by each term in the second binomial.
Combine like terms to simplify the expression.
Check your work by plugging in values for the variable to ensure that the expression is true.

Common Mistakes

Failing to use the distributive property when expanding and simplifying algebraic expressions.
Not combining like terms to simplify the expression.
Making errors when multiplying each term in the first binomial by each term in the second binomial.

Real-World Applications

Expanding and simplifying algebraic expressions has numerous real-world applications, including:

Solving equations and inequalities in physics, engineering, and economics.
Modeling population growth and decline in biology and ecology.
Analyzing data and making predictions in statistics and data analysis.

Final Thoughts

Expanding and simplifying algebraic expressions is a fundamental concept in mathematics, and it is essential for solving equations and inequalities. By following the step-by-step process outlined in this article, you will be able to expand and simplify polynomial expressions with ease. Remember to use the distributive property, combine like terms, and check your work to ensure that the expression is true. With practice and patience, you will become proficient in expanding and simplifying algebraic expressions and be able to apply this knowledge to real-world problems.

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Introduction

In our previous article, we explored the process of expanding and simplifying algebraic expressions, specifically the product of two binomials. We provided a step-by-step guide on how to expand and simplify the expression $(x-4)(x-6)$ using the distributive property and combining like terms. In this article, we will answer some of the most frequently asked questions about expanding and simplifying algebraic expressions.

Q&A

Q: What is the distributive property, and how is it used in expanding and simplifying algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , $a(b+c) = ab + ac$ . It is used to expand and simplify algebraic expressions by multiplying each term in the first binomial by each term in the second binomial.

Q: How do I apply the distributive property to expand and simplify algebraic expressions?

A: To apply the distributive property, you need to multiply each term in the first binomial by each term in the second binomial. For example, to expand the expression $(x-4)(x-6)$ , you would multiply each term in the first binomial, $(x-4)$ , by each term in the second binomial, $(x-6)$ .

Q: What is the difference between expanding and simplifying algebraic expressions?

A: Expanding an algebraic expression involves multiplying each term in the first binomial by each term in the second binomial, while simplifying an algebraic expression involves combining like terms to reduce the expression to its simplest form.

Q: How do I combine like terms to simplify an algebraic expression?

A: To combine like terms, you need to identify the terms that have the same variable and coefficient, and then add or subtract them. For example, in the expression $x^2 - 6x - 4x + 24$ , you would combine the like terms $-6x$ and $-4x$ to get $-10x$ .

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

A: Some common mistakes to avoid include failing to use the distributive property, not combining like terms, and making errors when multiplying each term in the first binomial by each term in the second binomial.

Q: How do I check my work when expanding and simplifying algebraic expressions?

A: To check your work, you can plug in values for the variable to ensure that the expression is true. For example, if you are expanding the expression $(x-4)(x-6)$ , you can plug in $x=0$ to get $(-4)(-6) = 24$ , which is true.

Q: What are some real-world applications of expanding and simplifying algebraic expressions?

A: Expanding and simplifying algebraic expressions has numerous real-world applications, including solving equations and inequalities in physics, engineering, and economics, modeling population growth and decline in biology and ecology, and analyzing data and making predictions in statistics and data analysis.

Tips and Tricks

Make sure to use the distributive property when expanding and simplifying algebraic expressions.
Combine like terms to simplify the expression.
Check your work by plugging in values for the variable to ensure that the expression is true.
Practice, practice, practice! The more you practice expanding and simplifying algebraic expressions, the more comfortable you will become with the process.

Common Mistakes

Failing to use the distributive property when expanding and simplifying algebraic expressions.
Not combining like terms to simplify the expression.
Making errors when multiplying each term in the first binomial by each term in the second binomial.

Real-World Applications

Expanding and simplifying algebraic expressions has numerous real-world applications, including:

Solving equations and inequalities in physics, engineering, and economics.
Modeling population growth and decline in biology and ecology.
Analyzing data and making predictions in statistics and data analysis.