Simplify The Expression: $ (b-c)(3d+e)-(b-c)(d-2e) $

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. It involves combining like terms, removing parentheses, and performing other operations to make the expression more manageable. In this article, we will simplify the given expression $(b-c)(3d+e)-(b-c)(d-2e)$ using various algebraic techniques.

Understanding the Expression

The given expression is a combination of two terms, each of which is a product of two binomials. The first term is $(b-c)(3d+e)$, and the second term is $(b-c)(d-2e)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$.

Applying the Distributive Property

To simplify the first term, we will apply the distributive property by multiplying each term in the first binomial $(b-c)$ with each term in the second binomial $(3d+e)$. This will give us:

$(b-c)(3d+e) = b(3d+e) - c(3d+e)$

Using the distributive property again, we can expand each term:

$b(3d+e) = 3bd + be$

$c(3d+e) = 3cd + ce$

Now, we can combine the two terms:

$3bd + be - 3cd - ce$

Simplifying the Second Term

To simplify the second term, we will apply the distributive property by multiplying each term in the first binomial $(b-c)$ with each term in the second binomial $(d-2e)$. This will give us:

$(b-c)(d-2e) = b(d-2e) - c(d-2e)$

Using the distributive property again, we can expand each term:

$b(d-2e) = bd - 2be$

$c(d-2e) = cd - 2ce$

Now, we can combine the two terms:

$bd - 2be - cd + 2ce$

Combining Like Terms

Now that we have simplified both terms, we can combine like terms to get the final expression. We will combine the terms with the same variable:

$3bd + be - 3cd - ce - bd + 2be + cd - 2ce$

Combining like terms, we get:

$2be + 3bd - 3cd - ce - 2ce$

Final Simplification

Now, we can simplify the expression further by combining like terms:

$2be + 3bd - 3cd - 3ce$

We can factor out the common term $-3$ from the last two terms:

$2be + 3bd - 3c(d + e)$

Conclusion

In this article, we simplified the given expression $(b-c)(3d+e)-(b-c)(d-2e)$ using the distributive property and combining like terms. The final simplified expression is $2be + 3bd - 3c(d + e)$. This expression is more manageable and easier to work with, making it a valuable tool for solving equations and inequalities.

Tips and Tricks

• When simplifying expressions, always look for opportunities to apply the distributive property.
• Combine like terms to make the expression more manageable.
• Factor out common terms to simplify the expression further.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

• Physics: Simplifying expressions is essential in physics to solve equations and inequalities related to motion, energy, and momentum.
• Engineering: Engineers use simplifying expressions to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Economists use simplifying expressions to model and analyze economic systems, including supply and demand curves.

Common Mistakes to Avoid

• Failing to apply the distributive property when simplifying expressions.
• Not combining like terms to make the expression more manageable.
• Factoring out common terms incorrectly.

Final Thoughts

Simplifying expressions is a fundamental skill in algebra that helps us solve equations and inequalities. By applying the distributive property, combining like terms, and factoring out common terms, we can simplify expressions and make them more manageable. With practice and patience, you can become proficient in simplifying expressions and tackle even the most complex problems.

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Introduction

In our previous article, we simplified the expression $(b-c)(3d+e)-(b-c)(d-2e)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional tips and tricks to help you master this skill.

Q&A

Q: What is the distributive property, and how is it used in simplifying 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 simplify expressions by multiplying each term in one binomial with each term in the other binomial.

Q: How do I apply the distributive property when simplifying expressions?

A: To apply the distributive property, simply multiply each term in one binomial with each term in the other binomial. For example, to simplify the expression $(b-c)(3d+e)$, you would multiply each term in the first binomial $(b-c)$ with each term in the second binomial $(3d+e)$.

Q: What is the difference between combining like terms and factoring out common terms?

A: Combining like terms involves adding or subtracting terms with the same variable, while factoring out common terms involves removing a common factor from each term. For example, in the expression $2be + 3bd - 3cd - 3ce$, we can combine like terms by adding the terms with the same variable, but we can also factor out the common term $-3$ from the last two terms.

Q: How do I know when to combine like terms and when to factor out common terms?

A: You should combine like terms when the terms have the same variable and the same exponent, and you should factor out common terms when there is a common factor that can be removed from each term.

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

A: Some common mistakes to avoid when simplifying expressions include failing to apply the distributive property, not combining like terms, and factoring out common terms incorrectly.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises in your algebra textbook or online resources. You can also try simplifying expressions on your own and checking your work with a calculator or a friend.

Tips and Tricks

• Always apply the distributive property when simplifying expressions.
• Combine like terms to make the expression more manageable.
• Factor out common terms to simplify the expression further.
• Use a calculator or a friend to check your work and ensure that you are simplifying expressions correctly.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

• Physics: Simplifying expressions is essential in physics to solve equations and inequalities related to motion, energy, and momentum.
• Engineering: Engineers use simplifying expressions to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Economists use simplifying expressions to model and analyze economic systems, including supply and demand curves.

Conclusion

Simplifying expressions is a fundamental skill in algebra that helps us solve equations and inequalities. By applying the distributive property, combining like terms, and factoring out common terms, we can simplify expressions and make them more manageable. With practice and patience, you can become proficient in simplifying expressions and tackle even the most complex problems.

Additional Resources

• Algebra textbooks and online resources
• Calculators and computer software
• Online communities and forums for algebra enthusiasts

Final Thoughts

Simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities. By mastering this skill, you can tackle even the most complex problems and become proficient in algebra. Remember to always apply the distributive property, combine like terms, and factor out common terms to simplify expressions and make them more manageable.