Which Rational Expression Has A Value Of 0 When $x=2$?A. $\frac{7x-5}{x^2+10}$B. $\frac{-3x+6}{8x-9}$C. $\frac{-5x+8}{x-2}$

Introduction

Rational expressions are a fundamental concept in algebra, and understanding how to solve them is crucial for success in mathematics. In this article, we will explore the concept of rational expressions and provide a step-by-step guide on how to solve them. We will also discuss the importance of rational expressions in real-world applications.

What are Rational Expressions?

A rational expression is a fraction that contains variables and constants in the numerator and denominator. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions. However, rational expressions can also be undefined, which means they have no value.

How to Solve Rational Expressions

To solve a rational expression, we need to follow these steps:

1. Simplify the expression: Simplify the numerator and denominator by combining like terms.
2. Factor the numerator and denominator: Factor the numerator and denominator to see if there are any common factors.
3. Cancel out common factors: Cancel out any common factors in the numerator and denominator.
4. Simplify the expression: Simplify the expression by combining like terms.

Example 1: Solving a Rational Expression

Let's consider the rational expression $\frac{7x-5}{x^2+10}$. To solve this expression, we need to follow the steps outlined above.

1. Simplify the expression: Simplify the numerator and denominator by combining like terms.
2. Factor the numerator and denominator: Factor the numerator and denominator to see if there are any common factors.
3. Cancel out common factors: Cancel out any common factors in the numerator and denominator.
4. Simplify the expression: Simplify the expression by combining like terms.

Which Rational Expression has a Value of 0 when $x=2$?

Now, let's consider the three rational expressions given in the problem:

A. $\frac{7x-5}{x^2+10}$ B. $\frac{-3x+6}{8x-9}$ C. $\frac{-5x+8}{x-2}$

To determine which rational expression has a value of 0 when $x=2$, we need to substitute $x=2$ into each expression and see which one equals 0.

Substituting $x=2$ into Expression A

Substituting $x=2$ into expression A, we get:

$\frac{7(2)-5}{(2)^2+10} = \frac{14-5}{4+10} = \frac{9}{14}$

Substituting $x=2$ into Expression B

Substituting $x=2$ into expression B, we get:

$\frac{-3(2)+6}{8(2)-9} = \frac{-6+6}{16-9} = \frac{0}{7}$

Substituting $x=2$ into Expression C

Substituting $x=2$ into expression C, we get:

$\frac{-5(2)+8}{(2)-2} = \frac{-10+8}{0}$

Conclusion

Based on the calculations above, we can see that expression B has a value of 0 when $x=2$. Therefore, the correct answer is:

B. $\frac{-3x+6}{8x-9}$

Real-World Applications of Rational Expressions

Rational expressions have numerous real-world applications, including:

• Engineering: Rational expressions are used to model and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Rational expressions are used to model and analyze economic systems, such as supply and demand curves.
• Computer Science: Rational expressions are used to model and analyze algorithms and data structures.

Conclusion

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Frequently Asked Questions about Rational Expressions

Q: What is a rational expression?

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

Q: How do I simplify a rational expression?

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

1. Simplify the numerator and denominator: Simplify the numerator and denominator by combining like terms.
2. Factor the numerator and denominator: Factor the numerator and denominator to see if there are any common factors.
3. Cancel out common factors: Cancel out any common factors in the numerator and denominator.
4. Simplify the expression: Simplify the expression by combining like terms.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as a fraction, such as 3/4 or 22/7. A rational expression, on the other hand, is a fraction that contains variables and constants in the numerator and denominator.

Q: How do I add and subtract rational expressions?

A: To add and subtract rational expressions, you need to follow these steps:

1. Find a common denominator: Find a common denominator for the two rational expressions.
2. Add or subtract the numerators: Add or subtract the numerators of the two rational expressions.
3. Simplify the expression: Simplify the expression by combining like terms.

Q: How do I multiply and divide rational expressions?

A: To multiply and divide rational expressions, you need to follow these steps:

1. Multiply or divide the numerators: Multiply or divide the numerators of the two rational expressions.
2. Multiply or divide the denominators: Multiply or divide the denominators of the two rational expressions.
3. Simplify the expression: Simplify the expression by combining like terms.

Q: What is the difference between a rational expression and a polynomial?

A: A polynomial is an expression that consists of variables and constants, such as 2x^2 + 3x - 4. A rational expression, on the other hand, is a fraction that contains variables and constants in the numerator and denominator.

Q: How do I solve rational expressions?

A: To solve rational expressions, you need to follow these steps:

1. Simplify the expression: Simplify the expression by combining like terms.
2. Factor the numerator and denominator: Factor the numerator and denominator to see if there are any common factors.
3. Cancel out common factors: Cancel out any common factors in the numerator and denominator.
4. Solve for the variable: Solve for the variable by setting the expression equal to 0.

Q: What is the importance of rational expressions in real-world applications?

A: Rational expressions have numerous real-world applications, including:

• Engineering: Rational expressions are used to model and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Rational expressions are used to model and analyze economic systems, such as supply and demand curves.
• Computer Science: Rational expressions are used to model and analyze algorithms and data structures.

Conclusion

In conclusion, rational expressions are a fundamental concept in algebra, and understanding how to solve them is crucial for success in mathematics. By following the steps outlined above, we can solve rational expressions and determine which one has a value of 0 when $x=2$. We also discussed the importance of rational expressions in real-world applications.