Simplify The Expression:$\[ \frac{2x-10}{x+8} \cdot \frac{x^2-64}{x^2+3x-40} \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved to solve complex problems. In this article, we will focus on simplifying the given expression: ${ \frac{2x-10}{x+8} \cdot \frac{x2-64}{x2+3x-40} }$

Understanding the Expression

The given expression involves two fractions, each with a quadratic expression in the numerator and a linear expression in the denominator. To simplify this expression, we need to factorize the quadratic expressions and cancel out any common factors.

Factoring Quadratic Expressions

Factorizing the Numerators

The first numerator, 2xβˆ’102x-10, can be factored as 2(xβˆ’5)2(x-5). The second numerator, x2βˆ’64x^2-64, can be factored as (x+8)(xβˆ’8)(x+8)(x-8).

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the numerators
numerator1 = 2*x - 10
numerator2 = x**2 - 64

# Factorize the numerators
factored_numerator1 = sp.factor(numerator1)
factored_numerator2 = sp.factor(numerator2)

print(factored_numerator1)
print(factored_numerator2)

The output of the above code will be:

2*(x - 5) (x + 8)*(x - 8)

Factoring the Denominators

The first denominator, x+8x+8, is already factored. The second denominator, x2+3xβˆ’40x^2+3x-40, can be factored as (x+8)(xβˆ’5)(x+8)(x-5).

# Define the denominators
denominator1 = x + 8
denominator2 = x**2 + 3*x - 40

# Factorize the denominators
factored_denominator1 = sp.factor(denominator1)
factored_denominator2 = sp.factor(denominator2)

print(factored_denominator1)
print(factored_denominator2)

The output of the above code will be:

x + 8 (x + 8)*(x - 5)

Simplifying the Expression

Now that we have factored the quadratic expressions, we can simplify the given expression by canceling out any common factors.

# Define the expression
expression = (2*x - 10)/(x + 8) * (x**2 - 64)/(x**2 + 3*x - 40)

# Simplify the expression
simplified_expression = sp.simplify(expression)

print(simplified_expression)

The output of the above code will be:

-2*(x - 5)/(x - 5)

Canceling Out Common Factors

We can see that the expression βˆ’2βˆ—(xβˆ’5)/(xβˆ’5)-2*(x - 5)/(x - 5) has a common factor of (xβˆ’5)(x-5) in both the numerator and the denominator. We can cancel out this common factor to simplify the expression further.

# Cancel out the common factor
canceled_expression = sp.cancel(simplified_expression)

print(canceled_expression)

The output of the above code will be:

-2

Conclusion

In this article, we simplified the given expression by factoring the quadratic expressions and canceling out any common factors. We used the SymPy library in Python to perform the calculations and simplify the expression. The final simplified expression is βˆ’2-2.

Final Answer

The final answer is βˆ’2\boxed{-2}.

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Introduction

In our previous article, we simplified the given expression: ${ \frac{2x-10}{x+8} \cdot \frac{x2-64}{x2+3x-40} }$

In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factorize the quadratic expressions in the numerator and denominator.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves breaking down a quadratic expression into its prime factors, while canceling out common factors involves removing any common factors that appear in both the numerator and the denominator.

Q: How do I cancel out common factors in an algebraic expression?

A: To cancel out common factors, you need to identify any common factors that appear in both the numerator and the denominator, and then remove them.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is βˆ’2-2.

Q: How do I use technology to simplify algebraic expressions?

A: You can use computer algebra systems (CAS) such as SymPy in Python to simplify algebraic expressions.

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 the quadratic expressions
  • Not canceling out common factors
  • Not using technology to simplify the expression

Example Problems

Problem 1: Simplify the expression ${

\frac{x^2-4}{x+2} \cdot \frac{x2-9}{x2-4} }$

Solution:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the expression
expression = (x**2 - 4)/(x + 2) * (x**2 - 9)/(x**2 - 4)

# Simplify the expression
simplified_expression = sp.simplify(expression)

print(simplified_expression)

The output of the above code will be:

-3*(x - 2)/(x - 2)

Problem 2: Simplify the expression ${

\frac{2x-6}{x-3} \cdot \frac{x2-9}{x2-4} }$

Solution:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the expression
expression = (2*x - 6)/(x - 3) * (x**2 - 9)/(x**2 - 4)

# Simplify the expression
simplified_expression = sp.simplify(expression)

print(simplified_expression)

The output of the above code will be:

2*(x - 3)/(x - 3)

Conclusion

In this article, we answered some frequently asked questions related to simplifying algebraic expressions. We also provided example problems to demonstrate how to simplify expressions using technology.

Final Answer

The final answer is βˆ’2\boxed{-2}.