Select The Correct Answer From Each Drop-down Menu.Consider This Expression:${ \frac{3}{x-2}+\frac{z-2}{z^2-4x+4} }$The Simplest Form Of The Expression Has $\square$ In The Numerator And $\square$ In The Denominator.

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given expression: ${ \frac{3}{x-2}+\frac{z-2}{z^2-4x+4} }$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the concepts involved.

Understanding the Expression

The given expression consists of two fractions: 3xβˆ’2\frac{3}{x-2} and zβˆ’2z2βˆ’4x+4\frac{z-2}{z^2-4x+4}. To simplify this expression, we need to find a common denominator and combine the fractions.

Common Denominator

The common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are (xβˆ’2)(x-2) and (z2βˆ’4x+4)(z^2-4x+4). To find the LCM, we need to factorize both expressions.

Factorizing the Denominators

The first denominator, (xβˆ’2)(x-2), is already factored. The second denominator, (z2βˆ’4x+4)(z^2-4x+4), can be factored as follows:

(z2βˆ’4x+4)=(zβˆ’2)2(z^2-4x+4) = (z-2)^2

Now that we have factored both denominators, we can find the LCM.

Finding the LCM

The LCM of (xβˆ’2)(x-2) and (zβˆ’2)2(z-2)^2 is (xβˆ’2)(zβˆ’2)2(x-2)(z-2)^2.

Simplifying the Expression

Now that we have found the common denominator, we can rewrite the expression with the common denominator:

3(zβˆ’2)2(xβˆ’2)(zβˆ’2)2+(zβˆ’2)(xβˆ’2)(xβˆ’2)(zβˆ’2)2\frac{3(z-2)^2}{(x-2)(z-2)^2} + \frac{(z-2)(x-2)}{(x-2)(z-2)^2}

We can simplify this expression by canceling out the common factors in the numerator and denominator.

Canceling Out Common Factors

The common factors in the numerator and denominator are (zβˆ’2)2(z-2)^2 and (xβˆ’2)(x-2). Canceling these factors, we get:

3(zβˆ’2)(zβˆ’2)+(xβˆ’2)(zβˆ’2)\frac{3(z-2)}{(z-2)} + \frac{(x-2)}{(z-2)}

Simplifying Further

We can simplify this expression further by canceling out the common factor (zβˆ’2)(z-2) in the numerator and denominator.

Final Simplified Expression

The final simplified expression is:

3+(xβˆ’2)(zβˆ’2)3 + \frac{(x-2)}{(z-2)}

Conclusion

In this article, we have simplified the given algebraic expression by finding a common denominator, combining the fractions, and canceling out common factors. The final simplified expression is 3+(xβˆ’2)(zβˆ’2)3 + \frac{(x-2)}{(z-2)}. This expression is in the simplest form, with 3\boxed{3} in the numerator and (zβˆ’2)\boxed{(z-2)} in the denominator.

Discussion

The process of simplifying algebraic expressions involves finding a common denominator, combining the fractions, and canceling out common factors. This process requires a clear understanding of the concepts involved, including factorization, LCM, and simplification. By following the steps outlined in this article, you can simplify complex algebraic expressions and arrive at the simplest form.

Example Problems

  1. Simplify the expression: ${ \frac{2}{x+1} + \frac{3}{x^2-1} }$
  2. Simplify the expression: ${ \frac{4}{z-2} + \frac{2}{z^2-4z+4} }$

Solutions

  1. The final simplified expression is: ${ \frac{2(x2-1)}{(x+1)(x2-1)} + \frac{3(x+1)}{(x+1)(x^2-1)} }$

2x2βˆ’2+3x+3(x+1)(x2βˆ’1)\frac{2x^2-2+3x+3}{(x+1)(x^2-1)}

2x2+3x+1(x+1)(xβˆ’1)\frac{2x^2+3x+1}{(x+1)(x-1)}

  1. The final simplified expression is: ${ \frac{4(z-2)}{(z-2)(z-2)^2} + \frac{2(z-2)2}{(z-2)(z-2)2} }$

4(zβˆ’2)(zβˆ’2)2+2(zβˆ’2)(zβˆ’2)2\frac{4(z-2)}{(z-2)^2} + \frac{2(z-2)}{(z-2)^2}

4(zβˆ’2)+2(zβˆ’2)(zβˆ’2)2\frac{4(z-2)+2(z-2)}{(z-2)^2}

6(zβˆ’2)(zβˆ’2)2\frac{6(z-2)}{(z-2)^2}

\frac{6}{z-2}$<br/> # Simplifying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given expression: ${ \frac{3}{x-2}+\frac{z-2}{z^2-4x+4} }$. We broke down the steps involved in simplifying this expression and provided a clear understanding of the concepts involved. In this article, we will answer some of the most 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 find a common denominator. This involves identifying the least common multiple (LCM) of the denominators of the fractions in the expression.

Q: How do I find the LCM of two expressions?

A: To find the LCM of two expressions, you need to factorize both expressions and then multiply the highest power of each factor that appears in either expression.

Q: What is the difference between a common denominator and a least common multiple?

A: A common denominator is the denominator that is common to all the fractions in an expression, while a least common multiple (LCM) is the smallest multiple that is common to all the denominators in an expression.

Q: Can I simplify an algebraic expression by canceling out common factors in the numerator and denominator?

A: Yes, you can simplify an algebraic expression by canceling out common factors in the numerator and denominator. However, you need to make sure that the factors you are canceling out are actually common to both the numerator and denominator.

Q: What is the final simplified expression for the given expression: ${

\frac{3}{x-2}+\frac{z-2}{z^2-4x+4} }$?

A: The final simplified expression for the given expression is: ${ 3 + \frac{(x-2)}{(z-2)} }

Q: How do I simplify an algebraic expression with multiple fractions?

A: To simplify an algebraic expression with multiple fractions, you need to find a common denominator for all the fractions and then combine them.

Q: Can I simplify an algebraic expression by combining like terms?

A: Yes, you can simplify an algebraic expression by combining like terms. However, you need to make sure that the terms you are combining are actually like terms.

Example Problems

  1. Simplify the expression: ${ \frac{2}{x+1} + \frac{3}{x^2-1} }$
  2. Simplify the expression: ${ \frac{4}{z-2} + \frac{2}{z^2-4z+4} }$

Solutions

  1. The final simplified expression is: ${ \frac{2(x2-1)}{(x+1)(x2-1)} + \frac{3(x+1)}{(x+1)(x^2-1)} }$

2x2βˆ’2+3x+3(x+1)(x2βˆ’1)\frac{2x^2-2+3x+3}{(x+1)(x^2-1)}

2x2+3x+1(x+1)(xβˆ’1)\frac{2x^2+3x+1}{(x+1)(x-1)}

  1. The final simplified expression is: ${ \frac{4(z-2)}{(z-2)(z-2)^2} + \frac{2(z-2)2}{(z-2)(z-2)2} }$

4(zβˆ’2)(zβˆ’2)2+2(zβˆ’2)(zβˆ’2)2\frac{4(z-2)}{(z-2)^2} + \frac{2(z-2)}{(z-2)^2}

4(zβˆ’2)+2(zβˆ’2)(zβˆ’2)2\frac{4(z-2)+2(z-2)}{(z-2)^2}

6(zβˆ’2)(zβˆ’2)2\frac{6(z-2)}{(z-2)^2}

6zβˆ’2\frac{6}{z-2}

Tips and Tricks

  1. Factorize the denominators: Factorizing the denominators can help you find the LCM and simplify the expression.
  2. Cancel out common factors: Canceling out common factors in the numerator and denominator can help you simplify the expression.
  3. Combine like terms: Combining like terms can help you simplify the expression.
  4. Check your work: Always check your work to make sure that the expression is simplified correctly.

Conclusion

Simplifying algebraic expressions is an important skill to master in mathematics. By following the steps outlined in this article and practicing with example problems, you can become proficient in simplifying algebraic expressions. Remember to factorize the denominators, cancel out common factors, combine like terms, and check your work to ensure that the expression is simplified correctly.