Expand And Simplify \left(m^2+2\right)\left(m^2+3\right)\left(m^2+6\right ].

Mar 13, 2025 by ADMIN 77 views

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Introduction

In this article, we will delve into the world of algebra and explore the process of expanding and simplifying a given algebraic expression. The expression in question is $\left(m^2+2\right)\left(m^2+3\right)\left(m^2+6\right)$ . We will use various algebraic techniques to simplify this expression and arrive at a more manageable form.

Understanding the Expression

Before we begin the process of expansion and simplification, let's take a closer look at the given expression. It consists of three quadratic terms, each in the form of $m^2 + c$ , where $c$ is a constant. The expression can be written as:

\left(m^2+2\right)\left(m^2+3\right)\left(m^2+6\right)

Expanding the Expression

To expand this expression, we will use the distributive property of multiplication over addition. This property states that for any real numbers $a$ , $b$ , and $c$ , we have:

a(b+c) = ab + ac

We can apply this property to each pair of terms in the given expression. Let's start by multiplying the first two terms:

\left(m^2+2\right)\left(m^2+3\right) = m^4 + 3m^2 + 2m^2 + 6

Using the distributive property again, we can simplify the expression further:

m^4 + 3m^2 + 2m^2 + 6 = m^4 + 5m^2 + 6

Simplifying the Expression

Now that we have expanded the first two terms, we can multiply the result by the third term:

\left(m^4 + 5m^2 + 6\right)\left(m^2+6\right)

Using the distributive property once again, we can expand this expression:

m^6 + 6m^4 + 5m^4 + 30m^2 + 6m^2 + 36

Combining like terms, we get:

m^6 + 11m^4 + 36m^2 + 36

Final Simplification

The expression is now in its simplest form. We can see that it consists of three terms: $m^6$ , $11m^4$ , and $36m^2 + 36$ . This is the final simplified form of the given algebraic expression.

Conclusion

In this article, we have expanded and simplified the given algebraic expression $\left(m^2+2\right)\left(m^2+3\right)\left(m^2+6\right)$ . We used the distributive property of multiplication over addition to expand the expression, and then combined like terms to simplify it further. The final simplified form of the expression is $m^6 + 11m^4 + 36m^2 + 36$ .

Additional Tips and Tricks

When expanding and simplifying algebraic expressions, it's essential to use the distributive property correctly.
Combining like terms is a crucial step in simplifying expressions.
Always check your work by plugging in simple values for the variables to ensure that the expression is correct.

Real-World Applications

The techniques used in this article have numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, the expansion and simplification of algebraic expressions are used to describe the motion of objects and the behavior of physical systems. In engineering, these techniques are used to design and optimize complex systems. In computer science, they are used to develop algorithms and solve problems efficiently.

Final Thoughts

Expanding and simplifying algebraic expressions is a fundamental skill that is essential in mathematics and many other fields. By mastering these techniques, you can solve complex problems and arrive at elegant solutions. Remember to always use the distributive property correctly, combine like terms, and check your work to ensure that the expression is correct. With practice and patience, you can become proficient in expanding and simplifying algebraic expressions and tackle even the most challenging problems with confidence.

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Introduction

In our previous article, we explored the process of expanding and simplifying a given algebraic expression. We used various algebraic techniques to simplify the expression $\left(m^2+2\right)\left(m^2+3\right)\left(m^2+6\right)$ and arrived at a more manageable form. In this article, we will address some common questions and concerns related to expanding and simplifying algebraic expressions.

Q&A

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

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$ , $b$ , and $c$ , we have:

a(b+c) = ab + ac

This property is used to expand algebraic expressions by multiplying each term in the first expression by each term in the second expression.

Q: How do I know when to combine like terms in an algebraic expression?

A: Like terms are terms that have the same variable raised to the same power. To combine like terms, you need to add or subtract the coefficients of the like terms. For example, in the expression $2x + 3x$ , the like terms are $2x$ and $3x$ . To combine them, you add the coefficients, resulting in $5x$ .

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

A: Expanding an algebraic expression involves multiplying each term in the first expression by each term in the second expression, using the distributive property. Simplifying an algebraic expression involves combining like terms and eliminating any unnecessary terms.

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

A: To check your work, you can plug in simple values for the variables and evaluate the expression. For example, if you have the expression $x^2 + 3x + 2$ , you can plug in $x = 1$ and evaluate the expression to get $1^2 + 3(1) + 2 = 6$ . If the result is correct, then your work is correct.

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 correctly
Failing to combine like terms
Introducing extraneous terms
Failing to check your work

Real-World Applications

Final Thoughts

Additional Resources

Khan Academy: Algebra
MIT OpenCourseWare: Algebra
Wolfram Alpha: Algebra Calculator

Conclusion

In this article, we have addressed some common questions and concerns related to expanding and simplifying algebraic expressions. We have also provided some real-world applications and additional resources for further learning. By mastering the techniques used in this article, you can become proficient in expanding and simplifying algebraic expressions and tackle even the most challenging problems with confidence.