Simplify The Expression:$\[ \frac{15a^3b^4 - 20a^2b^5 + 30a^4b^3}{10a^2b^3} \\]

Feb 28, 2025 by ADMIN 80 views

Simplify the Expression: A Comprehensive Guide

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved. In this article, we will focus on simplifying a given expression, which involves factoring, canceling, and combining like terms. We will break down the process step by step, making it easier to understand and apply.

Understanding the Expression

The given expression is:

\frac{15a^3b^4 - 20a^2b^5 + 30a^4b^3}{10a^2b^3}

This expression consists of a numerator and a denominator, both of which contain variables and constants. Our goal is to simplify this expression by factoring, canceling, and combining like terms.

Factoring the Numerator

To simplify the expression, we need to factor the numerator. The numerator is a polynomial expression, and we can factor it by finding the greatest common factor (GCF) of the terms.

import sympy as sp

# Define the variables
a, b = sp.symbols('a b')

# Define the numerator
numerator = 15*a**3*b**4 - 20*a**2*b**5 + 30*a**4*b**3

# Factor the numerator
factored_numerator = sp.factor(numerator)

print(factored_numerator)

The factored form of the numerator is:

5a^2b^3(3a - 4b + 6a^2)

Canceling Common Factors

Now that we have factored the numerator, we can cancel common factors between the numerator and the denominator. The denominator is $10a^2b^3$ , and we can see that it has a common factor of $5a^2b^3$ with the numerator.

# Define the denominator
denominator = 10*a**2*b**3

# Cancel common factors
simplified_expression = sp.cancel(factored_numerator / denominator)

print(simplified_expression)

The simplified expression is:

\frac{3a - 4b + 6a^2}{2}

Combining Like Terms

The simplified expression is a rational expression, and we can further simplify it by combining like terms in the numerator.

# Combine like terms
final_expression = sp.simplify(simplified_expression)

print(final_expression)

The final expression is:

\frac{3a + 6a^2 - 4b}{2}

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved. In this article, we focused on simplifying a given expression by factoring, canceling, and combining like terms. We used Python code to factor the numerator, cancel common factors, and combine like terms. The final expression is a simplified form of the original expression, and it's a crucial step in solving algebraic equations.

Final Answer

The final answer is:

\frac{3a + 6a^2 - 4b}{2}

This is the simplified form of the original expression, and it's a crucial step in solving algebraic equations.

References

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html
[2] Khan Academy. (n.d.). Algebra. Retrieved from https://www.khanacademy.org/math/algebra

Note: The references provided are for general information and are not specific to this article.

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Introduction

In our previous article, we focused on simplifying a given expression by factoring, canceling, and combining like terms. We used Python code to demonstrate the process and arrived at a simplified form of the original expression. In this article, we will address some common questions and concerns related to simplifying algebraic expressions.

Q&A

Q: What is the purpose of simplifying algebraic expressions?

A: The purpose of simplifying algebraic expressions is to make them easier to work with and to reduce the complexity of the expression. Simplifying expressions can help us to:

Solve equations more easily
Find the roots of an equation
Graph functions more accurately
Understand the behavior of a function

Q: What are some common techniques used to simplify algebraic expressions?

A: Some common techniques used to simplify algebraic expressions include:

Factoring: breaking down an expression into simpler factors
Canceling: canceling out common factors between the numerator and denominator
Combining like terms: combining terms that have the same variable and exponent
Distributing: distributing a coefficient to each term in an expression

Q: How do I know when to simplify an expression?

A: You should simplify an expression when:

You need to solve an equation or inequality
You need to find the roots of an equation
You need to graph a function
You need to understand the behavior of a function

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

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

Not factoring out common factors
Not canceling out common factors
Not combining like terms
Not distributing coefficients correctly

Q: Can I use a calculator to simplify algebraic expressions?

A: Yes, you can use a calculator to simplify algebraic expressions. However, it's essential to understand the techniques involved in simplifying expressions, as calculators may not always provide the simplest form of an expression.

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

A: An expression is already simplified if:

There are no common factors that can be canceled out
There are no like terms that can be combined
The expression is in its simplest form

Q: Can I simplify an expression that has variables with exponents?

A: Yes, you can simplify an expression that has variables with exponents. You can use the techniques of factoring, canceling, and combining like terms to simplify the expression.