Simplify The Expression: { (a-2)\left(a^2+2a+4\right)$}$

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Introduction

In this article, we will simplify the given expression $(a-2)\left(a^2+2a+4\right)$. This involves using the distributive property to multiply each term in the first expression by each term in the second expression. We will then combine like terms to simplify the expression.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b+c) = ab + ac$

This property can be extended to more than two terms, and it is a crucial tool for simplifying complex expressions.

Applying the Distributive Property

To simplify the given expression, we will apply the distributive property to each term in the first expression $(a-2)$ and multiply it by each term in the second expression $\left(a^2+2a+4\right)$.

$(a-2)\left(a^2+2a+4\right) = a\left(a^2+2a+4\right) - 2\left(a^2+2a+4\right)$

Multiplying Each Term

Now, we will multiply each term in the first expression by each term in the second expression.

$a\left(a^2+2a+4\right) = a^3 + 2a^2 + 4a$

$-2\left(a^2+2a+4\right) = -2a^2 - 4a - 8$

Combining Like Terms

Now, we will combine like terms to simplify the expression.

$a^3 + 2a^2 + 4a - 2a^2 - 4a - 8 = a^3 + 0a^2 + 0a - 8$

Simplifying the Expression

After combining like terms, we get:

$a^3 - 8$

This is the simplified expression.

Conclusion

In this article, we simplified the given expression $(a-2)\left(a^2+2a+4\right)$ using the distributive property and combining like terms. We started by applying the distributive property to each term in the first expression and multiplying it by each term in the second expression. We then multiplied each term and combined like terms to simplify the expression. The final simplified expression is $a^3 - 8$.

Frequently Asked Questions

• What is the distributive property? The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms.
• How do I apply the distributive property? To apply the distributive property, multiply each term in the first expression by each term in the second expression.
• What is the simplified expression? The simplified expression is $a^3 - 8$.

Step-by-Step Solution

1. Apply the distributive property to each term in the first expression and multiply it by each term in the second expression.
2. Multiply each term in the first expression by each term in the second expression.
3. Combine like terms to simplify the expression.
4. The final simplified expression is $a^3 - 8$.

Example Problems

• Simplify the expression $(a+3)\left(a^2-2a+1\right)$.
• Simplify the expression $(2x-1)\left(x^2+3x-2\right)$.

Tips and Tricks

• Make sure to apply the distributive property correctly to each term in the first expression and multiply it by each term in the second expression.
• Combine like terms carefully to simplify the expression.
• Check your work by plugging in values for the variables to ensure that the expression is simplified correctly.
  

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Introduction

In our previous article, we simplified the given expression $(a-2)\left(a^2+2a+4\right)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b+c) = ab + ac$

This property can be extended to more than two terms, and it is a crucial tool for simplifying complex expressions.

Q: How do I apply the distributive property?

A: To apply the distributive property, multiply each term in the first expression by each term in the second expression. For example, if we have the expression $(a-2)\left(a^2+2a+4\right)$, we would multiply each term in the first expression $(a-2)$ by each term in the second expression $\left(a^2+2a+4\right)$.

Q: What is the simplified expression?

A: The simplified expression is $a^3 - 8$. This is the result of applying the distributive property and combining like terms to the original expression $(a-2)\left(a^2+2a+4\right)$.

Q: Can I use the distributive property with more than two terms?

A: Yes, the distributive property can be extended to more than two terms. For example, if we have the expression $(a-2)\left(a^2+2a+4\right)+3\left(a^2+2a+4\right)$, we would apply the distributive property to each term in the first expression and multiply it by each term in the second expression, and then add the result to the third term.

Q: How do I know when to combine like terms?

A: You should combine like terms when you have two or more terms that have the same variable and exponent. For example, if we have the expression $a^2 + 2a^2 + 4a$, we would combine the like terms $a^2$ and $2a^2$ to get $3a^2$.

Q: Can I use the distributive property with negative numbers?

A: Yes, the distributive property can be used with negative numbers. For example, if we have the expression $(-a-2)\left(a^2+2a+4\right)$, we would apply the distributive property to each term in the first expression and multiply it by each term in the second expression.

Q: How do I check my work?

A: You should check your work by plugging in values for the variables to ensure that the expression is simplified correctly. For example, if we have the expression $a^3 - 8$, we could plug in the value $a=1$ to get $1^3 - 8 = -7$.

Tips and Tricks

• Make sure to apply the distributive property correctly to each term in the first expression and multiply it by each term in the second expression.
• Combine like terms carefully to simplify the expression.
• Check your work by plugging in values for the variables to ensure that the expression is simplified correctly.
• Use the distributive property with negative numbers and more than two terms.

Example Problems

• Simplify the expression $(a+3)\left(a^2-2a+1\right)$.
• Simplify the expression $(2x-1)\left(x^2+3x-2\right)$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $(a-2)\left(a^2+2a+4\right)$. We covered topics such as the distributive property, applying the distributive property, combining like terms, and checking work. We also provided tips and tricks for simplifying expressions and example problems for practice.