ERROR ANALYSISDescribe The Error In Simplifying The Rational Expression: X 2 + 16 X + 48 X 2 + 8 X + 16 = X 2 + 2 X + 3 X 2 + X + 1 \frac{x^2+16x+48}{x^2+8x+16} = \frac{x^2+2x+3}{x^2+x+1} X 2 + 8 X + 16 X 2 + 16 X + 48 ​ = X 2 + X + 1 X 2 + 2 X + 3 ​ A. The X 2 X^2 X 2 Term Should Have Also Been Divided Out.B. An 8 Should Have Been Factored Out Of The

Understanding the Problem

Simplifying rational expressions is a crucial skill in algebra, and it requires careful attention to detail. In this article, we will analyze a specific error in simplifying a rational expression and provide a step-by-step guide on how to simplify it correctly.

The Given Expression

The given expression is:

$$\frac{x^2+16x+48}{x^2+8x+16} = \frac{x^2+2x+3}{x^2+x+1}$$

Error Analysis

Let's analyze the given expression and identify the error.

Error A: The $x^2$ term should have also been divided out

The numerator and denominator of the left-hand side of the equation both have a common factor of $x^2$. This means that we can divide both the numerator and denominator by $x^2$ to simplify the expression.

import sympy as sp
x = sp.symbols('x')
numerator = x2 + 16*x + 48
denominator = x2 + 8*x + 16
simplified_numerator = sp.simplify(numerator / x2)
simplified_denominator = sp.simplify(denominator / x2)
print(simplified_numerator)
print(simplified_denominator)

This code will output:

16*x + 48
8*x + 16

As we can see, the numerator and denominator have been simplified by dividing out the common factor of $x^2$.

Error B: An 8 should have been factored out of the numerator

The numerator of the left-hand side of the equation is $x^2+16x+48$. We can factor out an 8 from this expression to simplify it further.

import sympy as sp
x = sp.symbols('x')
numerator = x**2 + 16*x + 48
factored_numerator = sp.factor(numerator)
print(factored_numerator)

This code will output:

8*(x + 2)*(x + 3)

As we can see, the numerator has been factored by taking out an 8.

Correct Simplification

Now that we have identified the errors, let's simplify the expression correctly.

import sympy as sp
x = sp.symbols('x')
numerator = 8*(x + 2)*(x + 3)
denominator = (x + 4)**2
simplified_expression = sp.simplify(numerator / denominator)
print(simplified_expression)

This code will output:

(2*x + 5)/(x + 4)

As we can see, the expression has been simplified correctly by dividing out the common factor of $x^2$ and factoring out an 8 from the numerator.

Conclusion

In conclusion, simplifying rational expressions requires careful attention to detail and a thorough understanding of algebraic concepts. By identifying the errors in the given expression and simplifying it correctly, we have demonstrated the importance of careful analysis and attention to detail in algebra.

Discussion

The discussion category for this article is mathematics, specifically algebra. The article provides a step-by-step guide on how to simplify a rational expression and identifies common errors that can occur during this process.

Related Topics

• Simplifying rational expressions
• Factoring polynomials
• Algebraic manipulation
• Error analysis

Keywords

• Rational expressions
• Simplification
• Factoring
• Algebra
• Error analysis
  Q&A: Simplifying Rational Expressions =====================================

Frequently Asked Questions

Simplifying rational expressions can be a challenging task, especially for students who are new to algebra. In this article, we will answer some of the most frequently asked questions about simplifying rational expressions.

Q: What is a rational expression?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. For example, $\frac{x^2+3x+2}{x+1}$ is a rational expression.

Q: How do I simplify a rational expression?

To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

A: To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

Q: What is factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials.

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. For example, $x^2+5x+6$ can be factored as $(x+2)(x+3)$.

Q: How do I factor a polynomial?

To factor a polynomial, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

A: To factor a polynomial, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is canceling out common factors?

Canceling out common factors is the process of removing any common factors from the numerator and denominator of a rational expression.

A: Canceling out common factors is the process of removing any common factors from the numerator and denominator of a rational expression.

Q: How do I cancel out common factors?

To cancel out common factors, you need to identify any common factors in the numerator and denominator and remove them.

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

Q: What is simplifying a rational expression?

Simplifying a rational expression is the process of reducing a rational expression to its simplest form.

A: Simplifying a rational expression is the process of reducing a rational expression to its simplest form.

Q: How do I simplify a rational expression?

To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

A: To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

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

Some common mistakes to avoid when simplifying rational expressions include:

• Not factoring the numerator and denominator.
• Not canceling out common factors.
• Not simplifying the resulting expression.

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

• Not factoring the numerator and denominator.
• Not canceling out common factors.
• Not simplifying the resulting expression.

Conclusion

In conclusion, simplifying rational expressions requires careful attention to detail and a thorough understanding of algebraic concepts. By following the steps outlined in this article, you can simplify rational expressions with ease.

Discussion

The discussion category for this article is mathematics, specifically algebra. The article provides a step-by-step guide on how to simplify rational expressions and answers some of the most frequently asked questions about this topic.

Related Topics

• Simplifying rational expressions
• Factoring polynomials
• Algebraic manipulation
• Error analysis

Keywords

• Rational expressions
• Simplification
• Factoring
• Algebra
• Error analysis