Simplify The Expression:$\[ \begin{array}{l} 3x^3 + 6x^2 + 9x - 15 \\ = \quad 5x^2 - 10x + 15 \end{array} \\]

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying a given algebraic expression, which involves combining like terms and rearranging the expression to its simplest form. We will use the given expression as an example and walk through the steps to simplify it.

Understanding the Given Expression

The given expression is:

$$\begin{array}{l} 3x^3 + 6x^2 + 9x - 15 \\ = \quad 5x^2 - 10x + 15 \end{array}$$

This expression consists of four terms: $3x^3$, $6x^2$, $9x$, and $-15$. The first three terms are polynomial expressions, while the last term is a constant.

Step 1: Identify Like Terms

Like terms are terms that have the same variable(s) raised to the same power. In this expression, we can identify the following like terms:

• $3x^3$ and no other like terms
• $6x^2$ and no other like terms
• $9x$ and $-10x$ are like terms because they both have the variable $x$ raised to the power of 1.

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them by adding or subtracting their coefficients. In this case, we can combine the like terms $9x$ and $-10x$ by adding their coefficients:

$$9x - 10x = -x$$

So, the expression now becomes:

$$3x^3 + 6x^2 - x - 15$$

Step 3: Rearrange the Expression

Now that we have combined the like terms, we can rearrange the expression to put it in a more simplified form. We can start by rearranging the terms in descending order of their exponents:

$$3x^3 + 6x^2 - x - 15$$

This is the simplified form of the given expression.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By identifying like terms and combining them, we can simplify complex expressions and make them easier to work with. In this article, we walked through the steps to simplify a given algebraic expression, which involved combining like terms and rearranging the expression to its simplest form.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Identify like terms: Like terms are terms that have the same variable(s) raised to the same power. Identify like terms by looking for terms with the same variable(s) raised to the same power.
• Combine like terms: Combine like terms by adding or subtracting their coefficients.
• Rearrange the expression: Rearrange the expression to put it in a more simplified form. Start by rearranging the terms in descending order of their exponents.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying algebraic expressions:

• Not identifying like terms: Failing to identify like terms can lead to incorrect simplifications.
• Not combining like terms: Failing to combine like terms can lead to incorrect simplifications.
• Not rearranging the expression: Failing to rearrange the expression can lead to incorrect simplifications.

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

• Science and engineering: Simplifying algebraic expressions is essential in science and engineering, where complex equations need to be solved to model real-world phenomena.
• Computer programming: Simplifying algebraic expressions is essential in computer programming, where complex algorithms need to be implemented to solve real-world problems.
• Finance: Simplifying algebraic expressions is essential in finance, where complex financial models need to be solved to make informed investment decisions.

Final Thoughts

Introduction

In our previous article, we walked through the steps to simplify a given algebraic expression. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in simplifying algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What is a like term?

A: A like term is a term that has the same variable(s) raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable(s) raised to the same power. For example, in the expression 3x^2 + 2x^2 + 5x, the like terms are 3x^2 and 2x^2 because they both have the variable x raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, add or subtract their coefficients. For example, in the expression 3x^2 + 2x^2 + 5x, the like terms 3x^2 and 2x^2 can be combined by adding their coefficients: 3x^2 + 2x^2 = 5x^2.

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 algebraic 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.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

1. Identify like terms.
2. Combine like terms by adding or subtracting their coefficients.
3. Rearrange the expression to put it in a more simplified form.
4. Check your work by plugging in a value for the variable and evaluating the expression.

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

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

• Not identifying like terms.
• Not combining like terms.
• Not rearranging the expression.
• Not following the order of operations.

Q: How do I know if an algebraic expression is simplified?

A: An algebraic expression is simplified when it cannot be simplified further. To check if an expression is simplified, try to combine any like terms and rearrange the expression to put it in a more simplified form.

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

A: Simplifying algebraic expressions has many real-world applications, including:

• Science and engineering: Simplifying algebraic expressions is essential in science and engineering, where complex equations need to be solved to model real-world phenomena.
• Computer programming: Simplifying algebraic expressions is essential in computer programming, where complex algorithms need to be implemented to solve real-world problems.
• Finance: Simplifying algebraic expressions is essential in finance, where complex financial models need to be solved to make informed investment decisions.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify complex expressions and make them easier to work with. Remember to identify like terms, combine like terms, and rearrange the expression to put it in a more simplified form. With practice and patience, you will become proficient in simplifying algebraic expressions and be able to tackle more complex expressions in the future.