Find $\left(3 E^2 - 5 E + 7\right)(6 E + 1$\].

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Introduction

In this article, we will explore the process of finding the product of two algebraic expressions, specifically $\left(3 e^2 - 5 e + 7\right)$ and $(6 e + 1)$. This involves using the distributive property and simplifying the resulting expression.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one expression by each term in another expression. In this case, we will apply the distributive property to each term in the first expression and multiply it by each term in the second expression.

Applying the Distributive Property

To find the product of $\left(3 e^2 - 5 e + 7\right)$ and $(6 e + 1)$, we will apply the distributive property as follows:

$\left(3 e^2 - 5 e + 7\right)(6 e + 1)$

$= 3 e^2 (6 e + 1) - 5 e (6 e + 1) + 7 (6 e + 1)$

Expanding Each Term

Now, we will expand each term in the expression using the distributive property.

Expanding the First Term

$3 e^2 (6 e + 1)$

$= 3 e^2 (6 e) + 3 e^2 (1)$

$= 18 e^3 + 3 e^2$

Expanding the Second Term

$- 5 e (6 e + 1)$

$= - 5 e (6 e) - 5 e (1)$

$= - 30 e^2 - 5 e$

Expanding the Third Term

$7 (6 e + 1)$

$= 7 (6 e) + 7 (1)$

$= 42 e + 7$

Combining Like Terms

Now, we will combine like terms in the expression.

$18 e^3 + 3 e^2 - 30 e^2 - 5 e + 42 e + 7$

$= 18 e^3 - 27 e^2 + 37 e + 7$

Conclusion

In this article, we have found the product of $\left(3 e^2 - 5 e + 7\right)$ and $(6 e + 1)$ using the distributive property and simplifying the resulting expression. The final answer is $\boxed{18 e^3 - 27 e^2 + 37 e + 7}$.

Final Answer

$\boxed{18 e^3 - 27 e^2 + 37 e + 7}$

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Introduction

In our previous article, we explored the process of finding the product of two algebraic expressions, specifically $\left(3 e^2 - 5 e + 7\right)$ and $(6 e + 1)$. We applied the distributive property and simplified the resulting expression to find the final answer. In this article, we will answer some common questions related to this topic.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one expression by each term in another expression.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term in one expression by each term in another expression. For example, if you have the expression $(a + b)(c + d)$, you would multiply each term in $(a + b)$ by each term in $(c + d)$.

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

A: Expanding an expression involves applying the distributive property to multiply each term in one expression by each term in another expression. Simplifying an expression involves combining like terms and removing any unnecessary parentheses or brackets.

Q: How do I combine like terms?

A: To combine like terms, you need to identify the terms that have the same variable and coefficient. For example, if you have the expression $2x + 3x$, you would combine the two terms to get $5x$.

Q: What is the final answer to the expression $\left(3 e^2 - 5 e + 7\right)(6 e + 1)$?

A: The final answer to the expression $\left(3 e^2 - 5 e + 7\right)(6 e + 1)$ is $\boxed{18 e^3 - 27 e^2 + 37 e + 7}$.

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

A: Yes, you can use the distributive property to multiply expressions with more than two terms. For example, if you have the expression $(a + b + c)(d + e + f)$, you would multiply each term in $(a + b + c)$ by each term in $(d + e + f)$.

Q: How do I know when to use the distributive property?

A: You should use the distributive property whenever you need to multiply expressions. This can help you to expand and simplify expressions, and to find the final answer.

Conclusion

In this article, we have answered some common questions related to the distributive property and how to apply it to multiply expressions. We have also provided examples and explanations to help you understand the concept better. If you have any more questions or need further clarification, please don't hesitate to ask.

Final Answer

$\boxed{18 e^3 - 27 e^2 + 37 e + 7}$