Simplify The Expression:$\[ \frac{(x-y)^2-z^2}{x^2-(y+z)^2} + \frac{(y-z)^2-x^2}{y^2-(z+x)^2} + \frac{(z-x)^2-y^2}{z^2-(x+y)^2} \\]

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Introduction

In this article, we will delve into the world of mathematics and explore a complex expression involving algebraic manipulations. The given expression consists of three fractions, each with a unique numerator and denominator. Our goal is to simplify this expression and uncover its underlying structure.

The Expression

The given expression is:

(xβˆ’y)2βˆ’z2x2βˆ’(y+z)2+(yβˆ’z)2βˆ’x2y2βˆ’(z+x)2+(zβˆ’x)2βˆ’y2z2βˆ’(x+y)2\frac{(x-y)^2-z^2}{x^2-(y+z)^2} + \frac{(y-z)^2-x^2}{y^2-(z+x)^2} + \frac{(z-x)^2-y^2}{z^2-(x+y)^2}

Algebraic Manipulations

To simplify the expression, we will employ various algebraic techniques, including factoring, expanding, and canceling. Our first step is to expand the numerators and denominators of each fraction.

Expanding the Numerators and Denominators

Let's start by expanding the first fraction:

(xβˆ’y)2βˆ’z2x2βˆ’(y+z)2=x2βˆ’2xy+y2βˆ’z2x2βˆ’y2βˆ’2yzβˆ’z2\frac{(x-y)^2-z^2}{x^2-(y+z)^2} = \frac{x^2-2xy+y^2-z^2}{x^2-y^2-2yz-z^2}

Similarly, we can expand the second and third fractions:

(yβˆ’z)2βˆ’x2y2βˆ’(z+x)2=y2βˆ’2yz+z2βˆ’x2y2βˆ’z2βˆ’2zxβˆ’x2\frac{(y-z)^2-x^2}{y^2-(z+x)^2} = \frac{y^2-2yz+z^2-x^2}{y^2-z^2-2zx-x^2}

(zβˆ’x)2βˆ’y2z2βˆ’(x+y)2=z2βˆ’2zx+x2βˆ’y2z2βˆ’x2βˆ’2xyβˆ’y2\frac{(z-x)^2-y^2}{z^2-(x+y)^2} = \frac{z^2-2zx+x^2-y^2}{z^2-x^2-2xy-y^2}

Factoring the Denominators

Now that we have expanded the numerators and denominators, we can factor the denominators to simplify the expression further.

Factoring the First Denominator

The first denominator can be factored as follows:

x2βˆ’y2βˆ’2yzβˆ’z2=(xβˆ’y)(x+y)βˆ’2yzβˆ’z2x^2-y^2-2yz-z^2 = (x-y)(x+y)-2yz-z^2

=(xβˆ’y)(x+y+z)= (x-y)(x+y+z)

Factoring the Second Denominator

The second denominator can be factored as follows:

y2βˆ’z2βˆ’2zxβˆ’x2=(yβˆ’z)(y+z)βˆ’2zxβˆ’x2y^2-z^2-2zx-x^2 = (y-z)(y+z)-2zx-x^2

=(yβˆ’z)(y+z+x)= (y-z)(y+z+x)

Factoring the Third Denominator

The third denominator can be factored as follows:

z2βˆ’x2βˆ’2xyβˆ’y2=(zβˆ’x)(z+x)βˆ’2xyβˆ’y2z^2-x^2-2xy-y^2 = (z-x)(z+x)-2xy-y^2

=(zβˆ’x)(z+x+y)= (z-x)(z+x+y)

Canceling Common Factors

Now that we have factored the denominators, we can cancel common factors between the numerators and denominators.

Canceling Common Factors in the First Fraction

The first fraction can be simplified by canceling the common factor (xβˆ’y)(x-y) between the numerator and denominator:

x2βˆ’2xy+y2βˆ’z2(xβˆ’y)(x+y+z)=x2βˆ’2xy+y2βˆ’z2x+y+z\frac{x^2-2xy+y^2-z^2}{(x-y)(x+y+z)} = \frac{x^2-2xy+y^2-z^2}{x+y+z}

Canceling Common Factors in the Second Fraction

The second fraction can be simplified by canceling the common factor (yβˆ’z)(y-z) between the numerator and denominator:

y2βˆ’2yz+z2βˆ’x2(yβˆ’z)(y+z+x)=y2βˆ’2yz+z2βˆ’x2y+z+x\frac{y^2-2yz+z^2-x^2}{(y-z)(y+z+x)} = \frac{y^2-2yz+z^2-x^2}{y+z+x}

Canceling Common Factors in the Third Fraction

The third fraction can be simplified by canceling the common factor (zβˆ’x)(z-x) between the numerator and denominator:

z2βˆ’2zx+x2βˆ’y2(zβˆ’x)(z+x+y)=z2βˆ’2zx+x2βˆ’y2z+x+y\frac{z^2-2zx+x^2-y^2}{(z-x)(z+x+y)} = \frac{z^2-2zx+x^2-y^2}{z+x+y}

Combining the Fractions

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

Combining the Fractions

The final expression can be obtained by adding the simplified fractions:

x2βˆ’2xy+y2βˆ’z2x+y+z+y2βˆ’2yz+z2βˆ’x2y+z+x+z2βˆ’2zx+x2βˆ’y2z+x+y\frac{x^2-2xy+y^2-z^2}{x+y+z} + \frac{y^2-2yz+z^2-x^2}{y+z+x} + \frac{z^2-2zx+x^2-y^2}{z+x+y}

Simplifying the Final Expression

The final expression can be simplified by combining like terms.

Simplifying the Final Expression

The final expression can be simplified as follows:

x2βˆ’2xy+y2βˆ’z2x+y+z+y2βˆ’2yz+z2βˆ’x2y+z+x+z2βˆ’2zx+x2βˆ’y2z+x+y\frac{x^2-2xy+y^2-z^2}{x+y+z} + \frac{y^2-2yz+z^2-x^2}{y+z+x} + \frac{z^2-2zx+x^2-y^2}{z+x+y}

=(x2βˆ’2xy+y2)βˆ’(z2βˆ’2zx+x2)x+y+z+(y2βˆ’2yz+z2)βˆ’(x2βˆ’2xy+y2)y+z+x+(z2βˆ’2zx+x2)βˆ’(y2βˆ’2yz+z2)z+x+y= \frac{(x^2-2xy+y^2)-(z^2-2zx+x^2)}{x+y+z} + \frac{(y^2-2yz+z^2)-(x^2-2xy+y^2)}{y+z+x} + \frac{(z^2-2zx+x^2)-(y^2-2yz+z^2)}{z+x+y}

=x2βˆ’2xy+y2βˆ’z2+2zxβˆ’x2x+y+z+y2βˆ’2yz+z2βˆ’x2+2xyβˆ’y2y+z+x+z2βˆ’2zx+x2βˆ’y2+2yzβˆ’z2z+x+y= \frac{x^2-2xy+y^2-z^2+2zx-x^2}{x+y+z} + \frac{y^2-2yz+z^2-x^2+2xy-y^2}{y+z+x} + \frac{z^2-2zx+x^2-y^2+2yz-z^2}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

=βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y= \frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}


# Simplify the Expression: A Mathematical Exploration - Q&A =====================================================

Introduction

In our previous article, we explored a complex expression involving algebraic manipulations. We simplified the expression by employing various algebraic techniques, including factoring, expanding, and canceling. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the original expression that we simplified?

A: The original expression is:

(xβˆ’y)2βˆ’z2x2βˆ’(y+z)2+(yβˆ’z)2βˆ’x2y2βˆ’(z+x)2+(zβˆ’x)2βˆ’y2z2βˆ’(x+y)2\frac{(x-y)^2-z^2}{x^2-(y+z)^2} + \frac{(y-z)^2-x^2}{y^2-(z+x)^2} + \frac{(z-x)^2-y^2}{z^2-(x+y)^2}

Q: What are the steps involved in simplifying the expression?

A: The steps involved in simplifying the expression are:

  1. Expanding the numerators and denominators of each fraction.
  2. Factoring the denominators to simplify the expression further.
  3. Canceling common factors between the numerators and denominators.
  4. Combining the simplified fractions to obtain the final expression.

Q: What is the final simplified expression?

A: The final simplified expression is:

βˆ’2xy+z2+2zxx+y+z+βˆ’x2+2xyβˆ’y2+2yzy+z+x+z2βˆ’2zx+y2+2yzz+x+y\frac{-2xy+z^2+2zx}{x+y+z} + \frac{-x^2+2xy-y^2+2yz}{y+z+x} + \frac{z^2-2zx+y^2+2yz}{z+x+y}

Q: Can you explain the concept of factoring in the context of simplifying the expression?

A: Factoring is a technique used to simplify an expression by breaking it down into its constituent parts. In the context of simplifying the expression, we factored the denominators to simplify the expression further.

Q: What is the significance of canceling common factors in simplifying the expression?

A: Canceling common factors is an important step in simplifying the expression. By canceling common factors, we can simplify the expression and make it easier to work with.

Q: Can you provide an example of how to simplify a similar expression?

A: Yes, here is an example of how to simplify a similar expression:

(xβˆ’y)2βˆ’z2x2βˆ’(y+z)2+(yβˆ’z)2βˆ’x2y2βˆ’(z+x)2+(zβˆ’x)2βˆ’y2z2βˆ’(x+y)2\frac{(x-y)^2-z^2}{x^2-(y+z)^2} + \frac{(y-z)^2-x^2}{y^2-(z+x)^2} + \frac{(z-x)^2-y^2}{z^2-(x+y)^2}

To simplify this expression, we can follow the same steps as before:

  1. Expand the numerators and denominators of each fraction.
  2. Factor the denominators to simplify the expression further.
  3. Cancel common factors between the numerators and denominators.
  4. Combine the simplified fractions to obtain the final expression.

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

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

  • Not expanding the numerators and denominators of each fraction.
  • Not factoring the denominators to simplify the expression further.
  • Not canceling common factors between the numerators and denominators.
  • Not combining the simplified fractions to obtain the final expression.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression. We provided examples and explanations to help clarify the concepts involved in simplifying the expression. By following the steps outlined in this article, you should be able to simplify similar expressions and become more confident in your ability to work with algebraic expressions.

Additional Resources

For more information on simplifying expressions, we recommend the following resources:

  • Khan Academy: Algebra
  • Mathway: Algebra
  • Wolfram Alpha: Algebra

These resources provide a wealth of information on algebra and can help you improve your skills in simplifying expressions.

Final Thoughts

Simplifying expressions is an important skill to have in mathematics. By following the steps outlined in this article, you can simplify complex expressions and become more confident in your ability to work with algebraic expressions. Remember to always expand the numerators and denominators of each fraction, factor the denominators to simplify the expression further, cancel common factors between the numerators and denominators, and combine the simplified fractions to obtain the final expression. With practice and patience, you can become proficient in simplifying expressions and tackle even the most complex algebraic problems.