Multiply And Simplify The Expression:$\[ (3u + 7)(6u^2 + 2u - 1) \\]

Introduction

In algebra, multiplying and simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. In this article, we will focus on multiplying and simplifying the given expression: $(3u + 7)(6u^2 + 2u - 1)$. We will break down the process into manageable steps, using the distributive property and combining like terms to simplify the expression.

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 powerful tool for multiplying and simplifying expressions.

Multiplying the Expression

To multiply the given expression, we will use the distributive property to expand the product of the two binomials. We will start by multiplying the first term of the first binomial, $3u$, by each term of the second binomial, $6u^2 + 2u - 1$. Then, we will multiply the second term of the first binomial, $7$, by each term of the second binomial.

Step 1: Multiply $3u$ by each term of the second binomial

$3u \cdot 6u^2 = 18u^3$ $3u \cdot 2u = 6u^2$ $3u \cdot (-1) = -3u$

Step 2: Multiply $7$ by each term of the second binomial

$7 \cdot 6u^2 = 42u^2$ $7 \cdot 2u = 14u$ $7 \cdot (-1) = -7$

Combining Like Terms

Now that we have multiplied each term of the first binomial by each term of the second binomial, we can combine like terms to simplify the expression. We will group together terms with the same variable and exponent.

Step 1: Combine like terms with $u^3$

There is only one term with $u^3$, which is $18u^3$.

Step 2: Combine like terms with $u^2$

We have two terms with $u^2$, which are $6u^2$ and $42u^2$. Combining these terms, we get:

$6u^2 + 42u^2 = 48u^2$

Step 3: Combine like terms with $u$

We have two terms with $u$, which are $-3u$ and $14u$. Combining these terms, we get:

$-3u + 14u = 11u$

Step 4: Combine like terms with the constant term

We have two terms with the constant term, which are $-7$ and $0$. Combining these terms, we get:

$-7 + 0 = -7$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by writing it in the form of a single polynomial.

$(3u + 7)(6u^2 + 2u - 1) = 18u^3 + 6u^2 - 3u + 42u^2 + 14u - 7$

Combining like terms, we get:

$18u^3 + 48u^2 + 11u - 7$

Conclusion

In this article, we multiplied and simplified the given expression using the distributive property and combining like terms. We broke down the process into manageable steps, using the distributive property to expand the product of the two binomials and combining like terms to simplify the expression. By following these steps, we were able to simplify the expression and write it in the form of a single polynomial.

Final Answer

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Introduction

In our previous article, we explored the process of multiplying and simplifying algebraic expressions using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about multiplying and simplifying algebraic expressions.

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 powerful tool for multiplying and simplifying expressions.

Q: How do I multiply two binomials using the distributive property?

A: To multiply two binomials using the distributive property, you need to multiply each term of the first binomial by each term of the second binomial. For example, to multiply $(3u + 7)(6u^2 + 2u - 1)$, you would multiply $3u$ by each term of the second binomial, and then multiply $7$ by each term of the second binomial.

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

A: Multiplying an expression involves using the distributive property to expand the product of two or more binomials. Simplifying an expression involves combining like terms to write the expression in the form of a single polynomial.

Q: How do I combine like terms?

A: To combine like terms, you need to group together terms with the same variable and exponent. For example, to combine the terms $6u^2$ and $42u^2$, you would add their coefficients to get $48u^2$.

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

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

• Forgetting to multiply each term of the first binomial by each term of the second binomial
• Not combining like terms
• Making errors when adding or subtracting coefficients
• Not checking the final answer for errors

Q: How can I practice multiplying and simplifying expressions?

A: You can practice multiplying and simplifying expressions by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and challenging yourself to simplify them.

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

A: Multiplying and simplifying expressions has many real-world applications, including:

• Solving equations and inequalities
• Modeling real-world situations using algebraic expressions
• Simplifying complex expressions in physics and engineering
• Creating and solving algebraic equations in computer science

Conclusion

In this article, we answered some frequently asked questions about multiplying and simplifying algebraic expressions. We covered topics such as the distributive property, multiplying binomials, combining like terms, and common mistakes to avoid. By following these tips and practicing regularly, you can become more confident and proficient in multiplying and simplifying expressions.

Final Tips

• Always check your work for errors
• Practice regularly to build your skills and confidence
• Use online resources and tools to help you with multiplying and simplifying expressions
• Challenge yourself to create and simplify your own expressions

Additional Resources

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

Final Answer

The final answer is: Practice makes perfect!