Simplify The Expression And Choose The Correct Answer:${ \frac{3x 2}{x 2-25} - \frac{x}{x+5} + \frac{2}{x-5}, \quad X \neq -5, 5 }$A. { \frac{2x^2+7x+5}{(x+5)(x-5)}$}$B. { \frac{2x^2+7x+10}{(x+5)(x-5)}$}$C.

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Introduction

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Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including factoring, fractions, and algebraic manipulation. In this article, we will simplify the given expression and choose the correct answer from the options provided.

The Given Expression

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The given expression is:

${ \frac{3x^2}{x^2-25} - \frac{x}{x+5} + \frac{2}{x-5}, \quad x \neq -5, 5 }$

This expression consists of three fractions, and our goal is to simplify it by combining the fractions and eliminating any common factors.

Step 1: Factor the Denominators

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The first step in simplifying the expression is to factor the denominators of each fraction. We can factor the expression $x^2-25$ as $(x+5)(x-5)$.

import sympy as sp
x = sp.symbols('x')

expression = x**2 - 25
factored_expression = sp.factor(expression)
print(factored_expression)

This will output: (x + 5)*(x - 5)

Step 2: Rewrite the Expression with Factored Denominators

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Now that we have factored the denominators, we can rewrite the expression with the factored denominators.

${ \frac{3x^2}{(x+5)(x-5)} - \frac{x}{x+5} + \frac{2}{x-5} }$

Step 3: Find a Common Denominator

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To combine the fractions, we need to find a common denominator. In this case, the common denominator is $(x+5)(x-5)$.

Step 4: Rewrite Each Fraction with the Common Denominator

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We can rewrite each fraction with the common denominator by multiplying the numerator and denominator of each fraction by the necessary factors.

${ \frac{3x^2}{(x+5)(x-5)} \cdot \frac{(x+5)}{(x+5)} - \frac{x}{x+5} \cdot \frac{(x-5)}{(x-5)} + \frac{2}{x-5} \cdot \frac{(x+5)}{(x+5)} }$

Step 5: Simplify the Expression

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Now that we have rewritten each fraction with the common denominator, we can simplify the expression by combining the fractions.

${ \frac{3x^2(x+5)}{(x+5)(x-5)} - \frac{x(x-5)}{(x+5)(x-5)} + \frac{2(x+5)}{(x+5)(x-5)} }$

Step 6: Combine the Fractions

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We can combine the fractions by adding and subtracting the numerators.

${ \frac{3x^2(x+5) - x(x-5) + 2(x+5)}{(x+5)(x-5)} }$

Step 7: Simplify the Numerator

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We can simplify the numerator by expanding and combining like terms.

${ \frac{3x^3 + 15x^2 - x^2 + 5x + 2x + 10}{(x+5)(x-5)} }$

Step 8: Combine Like Terms

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We can combine like terms in the numerator.

${ \frac{3x^3 + 14x^2 + 7x + 10}{(x+5)(x-5)} }$

Step 9: Factor the Numerator

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We can factor the numerator, but in this case, it is already in its simplest form.

Conclusion

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The simplified expression is:

${ \frac{3x^3 + 14x^2 + 7x + 10}{(x+5)(x-5)} }$

This expression is in the form of option B.

The final answer is: $\boxed{B}$

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Introduction

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In our previous article, we simplified the given expression and chose the correct answer from the options provided. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

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

A: The first step in simplifying the expression is to factor the denominators of each fraction. We can factor the expression $x^2-25$ as $(x+5)(x-5)$.

Q: Why do we need to find a common denominator?

A: We need to find a common denominator to combine the fractions. In this case, the common denominator is $(x+5)(x-5)$.

Q: How do we rewrite each fraction with the common denominator?

A: We can rewrite each fraction with the common denominator by multiplying the numerator and denominator of each fraction by the necessary factors.

Q: What is the next step after rewriting each fraction with the common denominator?

A: After rewriting each fraction with the common denominator, we can simplify the expression by combining the fractions.

Q: How do we combine the fractions?

A: We can combine the fractions by adding and subtracting the numerators.

Q: What is the final simplified expression?

A: The final simplified expression is:

${ \frac{3x^3 + 14x^2 + 7x + 10}{(x+5)(x-5)} }$

Q: Which option is the correct answer?

A: The correct answer is option B.

Q: What is the significance of the expression $x \neq -5, 5$?

A: The expression $x \neq -5, 5$ indicates that the given expression is undefined when $x$ is equal to $-5$ or $5$. This is because the denominators of the fractions are zero when $x$ is equal to $-5$ or $5$.

Q: How do we handle the expression when $x$ is equal to $-5$ or $5$?

A: When $x$ is equal to $-5$ or $5$, the expression is undefined. We can handle this by using limits or by simplifying the expression in a way that avoids the undefined values.

Q: What is the final answer?

A: The final answer is $\boxed{B}$.

Additional Tips and Tricks

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• When simplifying expressions, it's essential to factor the denominators and find a common denominator to combine the fractions.
• Use the distributive property to expand and combine like terms in the numerator.
• Factor the numerator if possible to simplify the expression further.
• Be careful when handling expressions with undefined values, and use limits or simplification techniques to avoid them.

Conclusion

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In this Q&A article, we provided additional information and clarification on the topic of simplifying expressions. We hope this article has been helpful in understanding the concept and providing a deeper insight into the topic.