Simplify The Expression: $2a^2b - 8ab + 6b$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us to manipulate and solve equations. In this article, we will focus on simplifying the given expression: $2a^2b - 8ab + 6b$. We will use various techniques such as factoring, combining like terms, and using the distributive property to simplify the expression.

Understanding the Expression

The given expression is a quadratic expression in two variables, $a$ and $b$. It consists of three terms: $2a^2b$, $-8ab$, and $6b$. To simplify the expression, we need to understand the properties of exponents and how to combine like terms.

Factoring Out the Greatest Common Factor (GCF)

The first step in simplifying the expression is to factor out the greatest common factor (GCF) of the three terms. The GCF of $2a^2b$, $-8ab$, and $6b$ is $2b$. We can factor out $2b$ from each term:

$$2a^2b - 8ab + 6b = 2b(a^2 - 4a + 3)$$

Using the Distributive Property

Now that we have factored out the GCF, we can use the distributive property to simplify the expression further. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can use this property to simplify the expression:

$$2b(a^2 - 4a + 3) = 2ba^2 - 8ba + 6b$$

Combining Like Terms

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: $-8ba$ and $6b$. We can combine these terms by adding their coefficients:

$$2ba^2 - 8ba + 6b = 2ba^2 - 2ba$$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression further. We can factor out the common factor $2b$ from the two terms:

$$2ba^2 - 2ba = 2b(a^2 - a)$$

Final Answer

The final simplified expression is $2b(a^2 - a)$. This is the simplest form of the given expression.

Conclusion

Simplifying expressions is an essential skill in algebra that helps us to manipulate and solve equations. In this article, we have used various techniques such as factoring, combining like terms, and using the distributive property to simplify the expression $2a^2b - 8ab + 6b$. We have factored out the greatest common factor, used the distributive property, and combined like terms to simplify the expression. The final simplified expression is $2b(a^2 - a)$.

Tips and Tricks

• When simplifying expressions, always look for the greatest common factor (GCF) and factor it out.
• Use the distributive property to simplify expressions with multiple terms.
• Combine like terms by adding their coefficients.
• Factor out common factors from the simplified expression.

Common Mistakes

• Failing to factor out the greatest common factor (GCF).
• Not using the distributive property to simplify expressions with multiple terms.
• Not combining like terms.
• Not factoring out common factors from the simplified expression.

Real-World Applications

Simplifying expressions has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, simplifying expressions is used to solve problems involving motion, energy, and momentum. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

Further Reading

For further reading on simplifying expressions, we recommend the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

References

• [1] Algebra: A Comprehensive Introduction, by Michael Artin
• [2] Calculus: Early Transcendentals, by James Stewart
• [3] Mathematics for Computer Science, by Eric Lehman and Tom Leighton
  Simplify the Expression: $2a^2b - 8ab + 6b$ - Q&A =====================================================

Introduction

In our previous article, we simplified the expression $2a^2b - 8ab + 6b$ using various techniques such as factoring, combining like terms, and using the distributive property. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the greatest common factor (GCF) of the three terms in the expression $2a^2b - 8ab + 6b$?

A: The greatest common factor (GCF) of the three terms is $2b$.

Q: How do I factor out the GCF from the expression $2a^2b - 8ab + 6b$?

A: To factor out the GCF, we need to divide each term by the GCF. In this case, we divide each term by $2b$:

$$2a^2b - 8ab + 6b = 2b(a^2 - 4a + 3)$$

Q: What is the distributive property, and how do I use it to simplify expressions?

A: The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can use this property to simplify expressions by distributing the coefficient to each term inside the parentheses.

Q: How do I combine like terms in the expression $2a^2b - 8ab + 6b$?

A: To combine like terms, we need to add the coefficients of the like terms. In this case, we have two like terms: $-8ab$ and $6b$. We can combine these terms by adding their coefficients:

$$2a^2b - 8ab + 6b = 2ba^2 - 2ba$$

Q: What is the final simplified expression of $2a^2b - 8ab + 6b$?

A: The final simplified expression is $2b(a^2 - a)$.

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

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

• Failing to factor out the greatest common factor (GCF).
• Not using the distributive property to simplify expressions with multiple terms.
• Not combining like terms.
• Not factoring out common factors from the simplified expression.

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

A: Simplifying expressions has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, simplifying expressions is used to solve problems involving motion, energy, and momentum. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

Q: Where can I find more resources on simplifying expressions?

A: For further reading on simplifying expressions, we recommend the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Conclusion

Simplifying expressions is an essential skill in algebra that helps us to manipulate and solve equations. In this article, we have answered some frequently asked questions (FAQs) related to simplifying expressions. We hope that this article has provided you with a better understanding of simplifying expressions and how to apply it in real-world scenarios.

Tips and Tricks

• When simplifying expressions, always look for the greatest common factor (GCF) and factor it out.
• Use the distributive property to simplify expressions with multiple terms.
• Combine like terms by adding their coefficients.
• Factor out common factors from the simplified expression.

Common Mistakes

• Failing to factor out the greatest common factor (GCF).
• Not using the distributive property to simplify expressions with multiple terms.
• Not combining like terms.
• Not factoring out common factors from the simplified expression.

Real-World Applications

Simplifying expressions has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, simplifying expressions is used to solve problems involving motion, energy, and momentum. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

Further Reading

For further reading on simplifying expressions, we recommend the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

References

• [1] Algebra: A Comprehensive Introduction, by Michael Artin
• [2] Calculus: Early Transcendentals, by James Stewart
• [3] Mathematics for Computer Science, by Eric Lehman and Tom Leighton