Write the simplified expression

Oct 13, 2025 by pinkandtrash.com 7 views

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. Simplifying polynomials is an essential skill in mathematics, as it helps to reduce complex expressions into their simplest form, making them easier to work with. In this article, we will explore the process of simplifying polynomials, using a specific example to illustrate the steps involved.

The Example Polynomial

The polynomial we will be simplifying is given by:

${ 10abc^2 + 23a^2bc - abc^2 - 15a^2bc + abc^2 - 2a^2bc }$

Step 1: Combine Like Terms

The first step in simplifying the polynomial is to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. In this case, we can combine the terms with the same variable(s) raised to the same power.

${ 10abc^2 + 23a^2bc - abc^2 - 15a^2bc + abc^2 - 2a^2bc }$

We can start by combining the terms with the variable $a$ raised to the power of 2:

${ 23a^2bc - 15a^2bc - 2a^2bc }$

Next, we can combine the terms with the variable $a$ raised to the power of 1:

${ 10abc^2 - abc^2 + abc^2 }$

Finally, we can combine the terms with the variable $a$ raised to the power of 0 (i.e., the constant terms):

${ 0 }$

Step 2: Simplify the Expression

Now that we have combined like terms, we can simplify the expression further by combining the remaining terms.

${ 10abc^2 + 23a^2bc - abc^2 - 15a^2bc + abc^2 - 2a^2bc }$

We can start by combining the terms with the variable $a$ raised to the power of 2:

${ 23a^2bc - 15a^2bc - 2a^2bc }$

This simplifies to:

${ 6a^2bc }$

Next, we can combine the terms with the variable $a$ raised to the power of 1:

${ 10abc^2 - abc^2 + abc^2 }$

This simplifies to:

${ 10abc^2 }$

Finally, we can combine the terms with the variable $a$ raised to the power of 0 (i.e., the constant terms):

${ 0 }$

Step 3: Write the Simplified Expression

Now that we have simplified the expression, we can write the final answer.

${ 6a^2bc + 10abc^2 }$

Conclusion

Simplifying polynomials is an essential skill in mathematics, as it helps to reduce complex expressions into their simplest form, making them easier to work with. By following the steps outlined in this article, we can simplify polynomials and write the final answer in a clear and concise manner.

Tips and Tricks

When simplifying polynomials, it's essential to combine like terms first.
Use the distributive property to expand expressions and combine like terms.
Simplify expressions by combining the remaining terms.
Use the order of operations (PEMDAS) to evaluate expressions.

Common Mistakes

Failing to combine like terms.
Not using the distributive property to expand expressions.
Not simplifying expressions by combining the remaining terms.
Not using the order of operations (PEMDAS) to evaluate expressions.

Real-World Applications

Simplifying polynomials has numerous real-world applications, including:

Engineering: Simplifying polynomials is essential in engineering, where complex expressions are used to model real-world systems.
Computer Science: Simplifying polynomials is used in computer science to optimize algorithms and improve performance.
Economics: Simplifying polynomials is used in economics to model economic systems and make predictions.

Final Answer

The final answer is:

${ 6a^2bc + 10abc^2 }$

Introduction

Simplifying polynomials is an essential skill in mathematics, as it helps to reduce complex expressions into their simplest form, making them easier to work with. In this article, we will answer some frequently asked questions about simplifying polynomials, providing a deeper understanding of the process and its applications.

Q: What is a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication. Polynomials can be written in the form:

${ ax^n + bx^{n-1} + cx^{n-2} + \ldots + k }$

where $a$ , $b$ , $c$ , $\ldots$ , $k$ are constants, and $x$ is the variable.

Q: What is the purpose of simplifying polynomials?

The purpose of simplifying polynomials is to reduce complex expressions into their simplest form, making them easier to work with. Simplifying polynomials helps to:

Reduce the number of terms in an expression
Eliminate unnecessary variables and coefficients
Make it easier to solve equations and inequalities
Improve the accuracy of calculations

Q: How do I simplify a polynomial?

To simplify a polynomial, follow these steps:

Combine like terms: Combine terms with the same variable(s) raised to the same power.
Simplify the expression: Simplify the expression by combining the remaining terms.
Write the simplified expression: Write the final answer in a clear and concise manner.

Q: What are like terms?

Like terms are terms that have the same variable(s) raised to the same power. For example:

${ 2x^2 + 3x^2 }$

are like terms, as they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

To combine like terms, follow these steps:

Identify the like terms: Identify the terms that have the same variable(s) raised to the same power.
Add or subtract the coefficients: Add or subtract the coefficients of the like terms.
Write the combined term: Write the combined term in the simplified expression.

Q: What is the distributive property?

The distributive property is a mathematical property that states:

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

The distributive property is used to expand expressions and combine like terms.

Q: How do I use the distributive property?

To use the distributive property, follow these steps:

Identify the expression: Identify the expression that needs to be expanded.
Apply the distributive property: Apply the distributive property to expand the expression.
Combine like terms: Combine like terms in the expanded expression.

Q: What is the order of operations (PEMDAS)?

The order of operations (PEMDAS) is a mathematical rule that states:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate exponents next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Evaluate addition and subtraction operations from left to right.

Q: How do I use the order of operations (PEMDAS)?

To use the order of operations (PEMDAS), follow these steps:

Evaluate expressions inside parentheses: Evaluate expressions inside parentheses first.
Evaluate exponents: Evaluate exponents next.
Evaluate multiplication and division: Evaluate multiplication and division operations from left to right.
Evaluate addition and subtraction: Evaluate addition and subtraction operations from left to right.

Conclusion

Simplifying polynomials is an essential skill in mathematics, as it helps to reduce complex expressions into their simplest form, making them easier to work with. By following the steps outlined in this article, you can simplify polynomials and write the final answer in a clear and concise manner.

Tips and Tricks

When simplifying polynomials, it's essential to combine like terms first.
Use the distributive property to expand expressions and combine like terms.
Simplify expressions by combining the remaining terms.
Use the order of operations (PEMDAS) to evaluate expressions.

Common Mistakes

Failing to combine like terms.
Not using the distributive property to expand expressions.
Not simplifying expressions by combining the remaining terms.
Not using the order of operations (PEMDAS) to evaluate expressions.

Real-World Applications

Simplifying polynomials has numerous real-world applications, including:

Engineering: Simplifying polynomials is essential in engineering, where complex expressions are used to model real-world systems.
Computer Science: Simplifying polynomials is used in computer science to optimize algorithms and improve performance.
Economics: Simplifying polynomials is used in economics to model economic systems and make predictions.

Final Answer

The final answer is:

${ 6a^2bc + 10abc^2 }$

This is the simplified form of the original polynomial expression.