Simplify The Expression: ${ \frac{x 3+x 2+x}{x 3-1}+\frac{2x+2}{1-x 3}+\frac{x+1}{x^2-2} }$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving complex problems and understanding mathematical concepts. The given expression, ${ \frac{x3+x2+x}{x3-1}+\frac{2x+2}{1-x3}+\frac{x+1}{x^2-2} }$, is a combination of three fractions, each with its own unique denominator. In this article, we will guide you through the process of simplifying this expression step by step, using algebraic manipulation techniques.

Factorization of Denominators

To simplify the given expression, we need to factorize the denominators of each fraction. The first fraction has a denominator of x3−1x^3-1, which can be factorized as (x−1)(x2+x+1)(x-1)(x^2+x+1). The second fraction has a denominator of 1−x31-x^3, which can be factorized as (1−x)(1+x+x2)(1-x)(1+x+x^2). The third fraction has a denominator of x2−2x^2-2, which can be factorized as (x−2)(x+2)(x-\sqrt{2})(x+\sqrt{2}).

# Factorization of Denominators

First Fraction


The denominator of the first fraction is x3−1x^3-1, which can be factorized as:
x3−1=(x−1)(x2+x+1)x^3-1 = (x-1)(x^2+x+1)


Second Fraction


The denominator of the second fraction is 1−x31-x^3, which can be factorized as:
1−x3=(1−x)(1+x+x2)1-x^3 = (1-x)(1+x+x^2)


Third Fraction


The denominator of the third fraction is x2−2x^2-2, which can be factorized as:
x2−2=(x−2)(x+2)x^2-2 = (x-\sqrt{2})(x+\sqrt{2})

Simplifying Each Fraction

Now that we have factorized the denominators, we can simplify each fraction individually.

First Fraction

The first fraction is x3+x2+x(x−1)(x2+x+1)\frac{x^3+x^2+x}{(x-1)(x^2+x+1)}. We can simplify this fraction by canceling out the common factor of xx in the numerator and denominator.

# Simplifying the First Fraction

Canceling Out Common Factors


The first fraction can be simplified as:
x3+x2+x(x−1)(x2+x+1)=x(x2+x+1)(x−1)(x2+x+1)\frac{x^3+x^2+x}{(x-1)(x^2+x+1)} = \frac{x(x^2+x+1)}{(x-1)(x^2+x+1)}


Canceling Out Common Factors (continued)


We can cancel out the common factor of x2+x+1x^2+x+1 in the numerator and denominator:
x(x2+x+1)(x−1)(x2+x+1)=xx−1\frac{x(x^2+x+1)}{(x-1)(x^2+x+1)} = \frac{x}{x-1}

Second Fraction

The second fraction is 2x+2(1−x)(1+x+x2)\frac{2x+2}{(1-x)(1+x+x^2)}. We can simplify this fraction by canceling out the common factor of 22 in the numerator and denominator.

# Simplifying the Second Fraction

Canceling Out Common Factors


The second fraction can be simplified as:
2x+2(1−x)(1+x+x2)=2(x+1)(1−x)(1+x+x2)\frac{2x+2}{(1-x)(1+x+x^2)} = \frac{2(x+1)}{(1-x)(1+x+x^2)}


Canceling Out Common Factors (continued)


We can cancel out the common factor of 22 in the numerator and denominator:
2(x+1)(1−x)(1+x+x2)=x+112(1−x)(1+x+x2)\frac{2(x+1)}{(1-x)(1+x+x^2)} = \frac{x+1}{\frac{1}{2}(1-x)(1+x+x^2)}


Simplifying the Denominator


We can simplify the denominator by multiplying both the numerator and denominator by 22:
x+112(1−x)(1+x+x2)=2(x+1)(1−x)(1+x+x2)\frac{x+1}{\frac{1}{2}(1-x)(1+x+x^2)} = \frac{2(x+1)}{(1-x)(1+x+x^2)}

Third Fraction

The third fraction is x+1(x−2)(x+2)\frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})}. We can simplify this fraction by canceling out the common factor of x+1x+1 in the numerator and denominator.

# Simplifying the Third Fraction

Canceling Out Common Factors


The third fraction can be simplified as:
x+1(x−2)(x+2)=x+1(x−2)(x+2)\frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})} = \frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})}

Combining the Fractions

Now that we have simplified each fraction individually, we can combine them to get the final simplified expression.

# Combining the Fractions

Adding the Fractions


The final simplified expression is:
xx−1+2(x+1)(1−x)(1+x+x2)+x+1(x−2)(x+2)\frac{x}{x-1} + \frac{2(x+1)}{(1-x)(1+x+x^2)} + \frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})}

Conclusion

In this article, we have guided you through the process of simplifying the given expression step by step, using algebraic manipulation techniques. We have factorized the denominators, simplified each fraction individually, and combined them to get the final simplified expression. We hope that this article has provided you with a comprehensive understanding of how to simplify expressions in algebra.

Final Answer

The final simplified expression is: xx−1+2(x+1)(1−x)(1+x+x2)+x+1(x−2)(x+2)\frac{x}{x-1} + \frac{2(x+1)}{(1-x)(1+x+x^2)} + \frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})}

Introduction

In our previous article, we guided you through the process of simplifying the given expression step by step, using algebraic manipulation techniques. In this article, we will answer some of the most frequently asked questions related to simplifying expressions in algebra.

Q&A

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

A: The first step in simplifying an expression is to factorize the denominators. This involves breaking down the denominator into its prime factors.

Q: How do I factorize a denominator?

A: To factorize a denominator, you need to find the prime factors of the denominator. For example, if the denominator is x3−1x^3-1, you can factorize it as (x−1)(x2+x+1)(x-1)(x^2+x+1).

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing the complexity of the expression by canceling out common factors, combining like terms, and performing other algebraic manipulations. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true.

Q: How do I simplify a fraction with a complex denominator?

A: To simplify a fraction with a complex denominator, you need to factorize the denominator and then cancel out any common factors between the numerator and denominator.

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

A: The final simplified expression for the given expression is: xx−1+2(x+1)(1−x)(1+x+x2)+x+1(x−2)(x+2)\frac{x}{x-1} + \frac{2(x+1)}{(1-x)(1+x+x^2)} + \frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})}

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

A: An expression is already simplified if there are no common factors between the numerator and denominator that can be canceled out.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. However, you need to be careful when canceling out common factors, as this can lead to an undefined expression.

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

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

  • Canceling out common factors between the numerator and denominator without checking if the expression is defined
  • Not checking if the expression is already simplified
  • Not using the correct order of operations when simplifying expressions

Conclusion

In this article, we have answered some of the most frequently asked questions related to simplifying expressions in algebra. We hope that this article has provided you with a comprehensive understanding of how to simplify expressions and avoid common mistakes.

Final Answer

The final simplified expression for the given expression is: xx−1+2(x+1)(1−x)(1+x+x2)+x+1(x−2)(x+2)\frac{x}{x-1} + \frac{2(x+1)}{(1-x)(1+x+x^2)} + \frac{x+1}{(x-\sqrt{2})(x+\sqrt{2})}