Simplify The Expression: (x+2)\left(3x^2+5x+10\right ]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. When we simplify an expression, we aim to rewrite it in a more compact and manageable form, often by combining like terms or factoring out common factors. In this article, we will focus on simplifying the given expression $(x+2)\left(3x^2+5x+10\right)$ using various algebraic techniques.

Understanding the Expression

Before we dive into simplifying the expression, let's break it down and understand its components. The given expression is a product of two binomials: $(x+2)$ and $\left(3x^2+5x+10\right)$. To simplify this expression, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one binomial by each term in the other binomial. In this case, we will multiply each term in $(x+2)$ by each term in $\left(3x^2+5x+10\right)$.

Step 1: Multiply the First Term

To simplify the expression, we will start by multiplying the first term in $(x+2)$, which is $x$, by each term in $\left(3x^2+5x+10\right)$. This will give us:

$x \cdot 3x^2 = 3x^3$ $x \cdot 5x = 5x^2$ $x \cdot 10 = 10x$

Step 2: Multiply the Second Term

Next, we will multiply the second term in $(x+2)$, which is $2$, by each term in $\left(3x^2+5x+10\right)$. This will give us:

$2 \cdot 3x^2 = 6x^2$ $2 \cdot 5x = 10x$ $2 \cdot 10 = 20$

Combining Like Terms

Now that we have multiplied each term in $(x+2)$ by each term in $\left(3x^2+5x+10\right)$, we can combine like terms to simplify the expression. Like terms are terms that have the same variable raised to the same power. In this case, we have:

$3x^3 + 5x^2 + 10x + 6x^2 + 10x + 20$

We can combine the like terms $5x^2$ and $6x^2$ to get $11x^2$, and the like terms $10x$ and $10x$ to get $20x$. The simplified expression is:

$3x^3 + 11x^2 + 20x + 20$

Final Answer

The final simplified expression is:

$3x^3 + 11x^2 + 20x + 20$

This expression cannot be simplified further using basic algebraic techniques. However, we can factor out a common factor of $x$ from the first three terms to get:

$x(3x^2 + 11x + 20) + 20$

This is the final simplified expression.

Conclusion

In this article, we simplified the given expression $(x+2)\left(3x^2+5x+10\right)$ using the distributive property and combining like terms. We started by multiplying each term in $(x+2)$ by each term in $\left(3x^2+5x+10\right)$, and then combined like terms to simplify the expression. The final simplified expression is $3x^3 + 11x^2 + 20x + 20$. This expression cannot be simplified further using basic algebraic techniques.

Frequently Asked Questions

• 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 binomial by each term in the other binomial.
• Q: How do I simplify an expression using the distributive property? A: To simplify an expression using the distributive property, multiply each term in one binomial by each term in the other binomial, and then combine like terms.
• Q: What are like terms? A: Like terms are terms that have the same variable raised to the same power.

Further Reading

• Algebraic Expressions: A Comprehensive Guide
• Simplifying Algebraic Expressions: Tips and Tricks
• Distributive Property: A Detailed Explanation
  

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Introduction

In our previous article, we simplified the expression $(x+2)\left(3x^2+5x+10\right)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions related to 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 expand expressions by multiplying each term in one binomial by each term in the other binomial. It states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, multiply each term in one binomial by each term in the other binomial, and then combine like terms. For example, to simplify the expression $(x+2)\left(3x^2+5x+10\right)$, we would multiply each term in $(x+2)$ by each term in $\left(3x^2+5x+10\right)$, and then combine like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, to combine the like terms $2x$ and $5x$, we would add their coefficients to get $7x$.

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different properties of algebra. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. The commutative property states that for any real numbers $a$ and $b$, $a+b = b+a$.

Q: How do I simplify an expression with multiple binomials?

A: To simplify an expression with multiple binomials, use the distributive property to multiply each term in one binomial by each term in the other binomials, and then combine like terms.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Conclusion

In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We discussed the distributive property, like terms, combining like terms, and the order of operations. We hope that this article has been helpful in clarifying any confusion you may have had about simplifying algebraic expressions.

Further Reading

• Algebraic Expressions: A Comprehensive Guide
• Simplifying Algebraic Expressions: Tips and Tricks
• Distributive Property: A Detailed Explanation
• Order of Operations: A Guide to Simplifying Expressions

Additional Resources

• Khan Academy: Algebra
• Mathway: Algebra Calculator
• Wolfram Alpha: Algebra Calculator

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics. By understanding the distributive property, like terms, and the order of operations, you can simplify complex expressions and solve equations with ease. Remember to always follow the order of operations and to combine like terms to simplify expressions. With practice and patience, you will become proficient in simplifying algebraic expressions and solving equations.