Choose The Correct Simplification Of 9 X 2 ( 4 X + 2 X 2 − 1 9x^2(4x + 2x^2 - 1 9 X 2 ( 4 X + 2 X 2 − 1 ].A. 18 X 4 + 36 X 3 − 9 X 2 18x^4 + 36x^3 - 9x^2 18 X 4 + 36 X 3 − 9 X 2 B. 18 X 4 − 36 X 3 + 9 X 2 18x^4 - 36x^3 + 9x^2 18 X 4 − 36 X 3 + 9 X 2 C. 36 X 4 + 18 X 3 − 9 X 2 36x^4 + 18x^3 - 9x^2 36 X 4 + 18 X 3 − 9 X 2 D. 36 X 4 − 13 X 3 + 9 X 2 36x^4 - 13x^3 + 9x^2 36 X 4 − 13 X 3 + 9 X 2

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific algebraic expression, $9x^2(4x + 2x^2 - 1)$, and explore the correct simplification among the given options.

Understanding the Expression

The given expression is $9x^2(4x + 2x^2 - 1)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. In this case, we have $9x^2$ as the factor, and $(4x + 2x^2 - 1)$ as the expression inside the parentheses.

Step 1: Apply the Distributive Property

To simplify the expression, we will apply the distributive property by multiplying $9x^2$ with each term inside the parentheses.

import sympy as sp
x = sp.symbols('x')

expr = 9x**2(4x + 2x**2 - 1)

simplified_expr = sp.expand(expr)

Step 2: Simplify the Expression

After applying the distributive property, we get:

$$9x^2(4x + 2x^2 - 1) = 36x^3 + 18x^4 - 9x^2$$

Comparing with the Options

Now, let's compare the simplified expression with the given options:

A. $18x^4 + 36x^3 - 9x^2$ B. $18x^4 - 36x^3 + 9x^2$ C. $36x^4 + 18x^3 - 9x^2$ D. $36x^4 - 13x^3 + 9x^2$

Conclusion

Based on the simplified expression, we can conclude that the correct answer is:

A. $18x^4 + 36x^3 - 9x^2$

This is because the simplified expression matches option A exactly.

Tips and Tricks

When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS):

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.

By following these steps and applying the distributive property, you can simplify even the most complex algebraic expressions.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

1. Not applying the distributive property: Failing to apply the distributive property can lead to incorrect simplifications.
2. Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect simplifications.
3. Not simplifying like terms: Failing to simplify like terms can lead to incorrect simplifications.

By avoiding these common mistakes, you can ensure that your algebraic expressions are simplified correctly.

Conclusion

Introduction

In our previous article, we explored the concept of simplifying algebraic expressions, focusing on the expression $9x^2(4x + 2x^2 - 1)$. We applied the distributive property and simplified the expression to arrive at the correct answer. In this article, we will address some common questions and concerns related to simplifying algebraic expressions.

Q: What is the distributive property, and how is it used in simplifying algebraic expressions?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the factor outside the parentheses. In the expression $9x^2(4x + 2x^2 - 1)$, we applied the distributive property by multiplying $9x^2$ with each term inside the parentheses: $4x$, $2x^2$, and $-1$.

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

A: You should apply the distributive property whenever you have a factor outside the parentheses and an expression inside the parentheses. This is a common scenario in algebraic expressions, and applying the distributive property will help you simplify the expression.

Q: What is the order of operations (PEMDAS), and how does it relate to simplifying algebraic expressions?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which we perform mathematical operations. The acronym PEMDAS stands for:

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.

When simplifying algebraic expressions, it's essential to follow the order of operations to ensure that you perform the operations in the correct order.

Q: How do I simplify like terms in an algebraic expression?

A: Like terms are terms that have the same variable and exponent. To simplify like terms, you can combine them by adding or subtracting their coefficients. For example, in the expression $2x^2 + 3x^2$, the like terms are $2x^2$ and $3x^2$. You can combine them by adding their coefficients: $2x^2 + 3x^2 = 5x^2$.

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

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

1. Not applying the distributive property: Failing to apply the distributive property can lead to incorrect simplifications.
2. Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect simplifications.
3. Not simplifying like terms: Failing to simplify like terms can lead to incorrect simplifications.

By avoiding these common mistakes, you can ensure that your algebraic expressions are simplified correctly.

Q: How can I practice simplifying algebraic expressions?

A: There are many resources available to help you practice simplifying algebraic expressions, including:

1. Online practice problems: Websites like Khan Academy, Mathway, and IXL offer practice problems and exercises to help you practice simplifying algebraic expressions.
2. Textbooks and workbooks: Many algebra textbooks and workbooks include practice problems and exercises to help you practice simplifying algebraic expressions.
3. Math apps: Math apps like Photomath and Math Tricks offer interactive practice problems and exercises to help you practice simplifying algebraic expressions.

By practicing regularly, you can become proficient in simplifying algebraic expressions and build your confidence in algebra.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for students and professionals alike. By applying the distributive property, following the order of operations (PEMDAS), and simplifying like terms, you can simplify even the most complex algebraic expressions. Remember to avoid common mistakes, such as not applying the distributive property, not following the order of operations, and not simplifying like terms. With practice and patience, you can become proficient in simplifying algebraic expressions.