Simplify: { (x-5)\left(x^2-2x-6\right)$}$

Mar 11, 2025 by ADMIN 42 views

$Simplify: ${$(x-5)\left(x^2-2x-6\right)\$}$$

Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. One common technique for simplifying expressions is to use the distributive property, which allows us to multiply each term in one expression by each term in another expression. In this article, we will use the distributive property to simplify the given expression: ${$ (x-5)\left(x^2-2x-6\right)$}$. We will also explore other techniques for simplifying expressions and provide examples to illustrate these techniques.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply each term in one expression by each term in another expression. The distributive property can be written as:

a(b + c) = ab + ac

This means that we can multiply each term in the expression (b + c) by the term a, and the result will be the sum of the products of each term in (b + c) with a.

Applying the Distributive Property

To simplify the given expression, we will apply the distributive property by multiplying each term in the expression (x - 5) by each term in the expression (x^2 - 2x - 6).

${$ (x-5)\left(x^2-2x-6\right)$ = (x-5)(x^2) - (x-5)(2x) - (x-5)(6)}$

Expanding the Terms

Now, we will expand each term in the expression by multiplying each term in (x - 5) by each term in (x^2 - 2x - 6).

${$ (x-5)(x^2) = x^3 - 5x^2}$

${$ (x-5)(2x) = 2x^2 - 10x}$

${$ (x-5)(6) = 6x - 30}$

Combining Like Terms

Now, we will combine like terms in the expression.

${$ x^3 - 5x^2 - 2x^2 + 10x + 6x - 30}$

Simplifying the Expression

Finally, we will simplify the expression by combining like terms.

${$ x^3 - 7x^2 + 16x - 30}$

Conclusion

In this article, we used the distributive property to simplify the given expression: ${$ (x-5)\left(x^2-2x-6\right)$}$. We also explored other techniques for simplifying expressions and provided examples to illustrate these techniques. By applying the distributive property and combining like terms, we were able to simplify the expression and arrive at the final result.

Additional Techniques for Simplifying Expressions

In addition to the distributive property, there are several other techniques for simplifying expressions. Some of these techniques include:

Factoring

Factoring involves expressing an expression as a product of simpler expressions. For example, the expression x^2 + 5x + 6 can be factored as (x + 3)(x + 2).

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable and exponent. For example, the expression 2x^2 + 3x^2 can be combined as 5x^2.

Canceling Common Factors

Canceling common factors involves canceling out common factors in two or more expressions. For example, the expression (x + 2)(x + 3) can be canceled as x + 2.

Examples of Simplifying Expressions

Here are some examples of simplifying expressions using the distributive property and other techniques:

Example 1

Simplify the expression: ${$ (x+2)\left(x^2-3x-4\right)$}$

Using the distributive property, we can multiply each term in the expression (x + 2) by each term in the expression (x^2 - 3x - 4).

${$ (x+2)\left(x^2-3x-4\right)$ = (x+2)(x^2) - (x+2)(3x) - (x+2)(4)}$

Expanding the terms, we get:

${$ x^3 + 2x^2 - 3x^2 - 6x - 4x - 8}$

Combining like terms, we get:

${$ x^3 - x^2 - 10x - 8}$

Example 2

Simplify the expression: ${$ (x-3)\left(x^2+2x+1\right)$}$

Using the distributive property, we can multiply each term in the expression (x - 3) by each term in the expression (x^2 + 2x + 1).

${$ (x-3)\left(x^2+2x+1\right)$ = (x-3)(x^2) + (x-3)(2x) + (x-3)(1)}$

Expanding the terms, we get:

${$ x^3 - 3x^2 + 2x^2 - 6x + x - 3}$

Combining like terms, we get:

${$ x^3 - 2x^2 - 5x - 3}$

Conclusion

Introduction

In our previous article, we used the distributive property to simplify the given expression: ${$ (x-5)\left(x^2-2x-6\right)$}$. We also explored other techniques for simplifying expressions and provided examples to illustrate these techniques. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply each term in one expression by each term in another expression. The distributive property can be written as:

a(b + c) = ab + ac

This means that we can multiply each term in the expression (b + c) by the term a, and the result will be the sum of the products of each term in (b + c) with a.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, you need to multiply each term in one expression by each term in another expression. For example, to simplify the expression (x + 2)(x^2 - 3x - 4), you would multiply each term in (x + 2) by each term in (x^2 - 3x - 4).

Q: What is the difference between factoring and simplifying an expression?

A: Factoring involves expressing an expression as a product of simpler expressions. For example, the expression x^2 + 5x + 6 can be factored as (x + 3)(x + 2). Simplifying an expression, on the other hand, involves combining like terms and eliminating any unnecessary terms.

Q: How do I combine like terms in an expression?

A: To combine like terms, you need to add or subtract terms that have the same variable and exponent. For example, the expression 2x^2 + 3x^2 can be combined as 5x^2.

Q: What is the difference between canceling common factors and simplifying an expression?

A: Canceling common factors involves canceling out common factors in two or more expressions. For example, the expression (x + 2)(x + 3) can be canceled as x + 2. Simplifying an expression, on the other hand, involves combining like terms and eliminating any unnecessary terms.

Q: How do I know when to use the distributive property and when to use other techniques for simplifying expressions?

A: The distributive property is a powerful tool for simplifying expressions, but it is not always the best approach. If you have an expression that can be factored, it is often better to factor it first. If you have an expression that can be combined using like terms, it is often better to combine those terms first.

Q: Can I simplify an expression that has multiple variables?

A: Yes, you can simplify an expression that has multiple variables. The distributive property and other techniques for simplifying expressions can be applied to expressions with multiple variables.

Q: How do I check my work when simplifying an expression?

A: To check your work, you can plug in a value for the variable and see if the expression simplifies to the expected result. You can also use a calculator or computer program to check your work.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions. We covered topics such as the distributive property, factoring, combining like terms, canceling common factors, and checking work. By understanding these concepts and techniques, you can simplify expressions with confidence and accuracy.

Additional Resources

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

Algebra textbooks: There are many algebra textbooks available that cover the topics of simplifying expressions and other algebra concepts.
Online resources: There are many online resources available that provide tutorials, examples, and practice problems for simplifying expressions.
Calculators and computer programs: Calculators and computer programs can be used to check your work and simplify expressions.

Final Thoughts

Simplifying expressions is an important skill in algebra, and it can be used to solve a wide range of problems. By understanding the distributive property and other techniques for simplifying expressions, you can simplify expressions with confidence and accuracy. Remember to always check your work and use a calculator or computer program to check your answers. With practice and patience, you can become proficient in simplifying expressions and solving algebra problems.