Simplify the second fraction

Oct 13, 2025 by windailys.com 20 views

$Simplify the second fraction$

Understanding Rational Expressions

Rational expressions are fractions that contain variables and constants in the numerator and denominator. They can be simplified by factoring the numerator and denominator, canceling out common factors, and then combining the remaining terms. In this article, we will explore Jeremy's error in simplifying a rational expression and provide the correct result.

The Expression

Jeremy was given the expression $\frac{2x^2 - 10}{x - 5} - \frac{x^2 + 15}{x - 5}$ . His teacher told him that his answer was incorrect, and we need to identify the error in his simplification process.

Jeremy's Simplification Process

Jeremy's simplification process can be broken down into the following steps:

Factor the numerator: Jeremy factored the numerator of the first fraction as $2x^2 - 10 = 2(x^2 - 5)$ .
Cancel out common factors: Jeremy canceled out the common factor $(x - 5)$ between the numerator and denominator of the first fraction.
Simplify the second fraction: Jeremy left the second fraction as is, without simplifying it.
Subtract the fractions: Jeremy subtracted the two fractions, resulting in $\frac{2(x^2 - 5)}{x - 5} - \frac{x^2 + 15}{x - 5}$ .

Error in Jeremy's Simplification Process

Jeremy's error lies in the fact that he canceled out the common factor $(x - 5)$ between the numerator and denominator of the first fraction without considering the second fraction. When subtracting fractions, we need to have a common denominator, which in this case is $(x - 5)$ . However, Jeremy did not simplify the second fraction, which resulted in an incorrect answer.

Correct Simplification Process

To simplify the expression correctly, we need to follow these steps:

Factor the numerator: Factor the numerator of the first fraction as $2x^2 - 10 = 2(x^2 - 5)$ .
Simplify the second fraction: Simplify the second fraction by factoring the numerator as $x^2 + 15 = (x + \sqrt{15})(x - \sqrt{15})$ .
Find a common denominator: The common denominator is $(x - 5)$ .
Subtract the fractions: Subtract the two fractions, resulting in $\frac{2(x^2 - 5)}{x - 5} - \frac{(x + \sqrt{15})(x - \sqrt{15})}{x - 5}$ .
Combine the fractions: Combine the two fractions by subtracting the numerators, resulting in $\frac{2(x^2 - 5) - (x + \sqrt{15})(x - \sqrt{15})}{x - 5}$ .

Final Answer

The final answer is $\frac{2(x^2 - 5) - (x + \sqrt{15})(x - \sqrt{15})}{x - 5}$ . This expression cannot be simplified further without knowing the value of $x$ .

Conclusion

Q: What is the main error in Jeremy's simplification process?

A: The main error in Jeremy's simplification process is that he canceled out the common factor $(x - 5)$ between the numerator and denominator of the first fraction without considering the second fraction.

Q: Why is it necessary to have a common denominator when subtracting fractions?

A: It is necessary to have a common denominator when subtracting fractions because it allows us to subtract the numerators directly. Without a common denominator, the fractions cannot be subtracted.

Q: How do I simplify a rational expression with multiple terms in the numerator?

A: To simplify a rational expression with multiple terms in the numerator, you need to factor the numerator and denominator, cancel out common factors, and then combine the remaining terms.

Q: What is the correct order of operations when simplifying a rational expression?

A: The correct order of operations when simplifying a rational expression is:

Factor the numerator and denominator.
Cancel out common factors.
Combine the remaining terms.

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

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful not to cancel out the variable with a term in the numerator.

Q: How do I know when a rational expression cannot be simplified further?

A: A rational expression cannot be simplified further when there are no common factors between the numerator and denominator, or when the expression is already in its simplest form.

Q: What is the final answer to the expression $\frac{2x^2 - 10}{x - 5} - \frac{x^2 + 15}{x - 5}$ ?

A: The final answer to the expression $\frac{2x^2 - 10}{x - 5} - \frac{x^2 + 15}{x - 5}$ is $\frac{2(x^2 - 5) - (x + \sqrt{15})(x - \sqrt{15})}{x - 5}$ .

Q: Can I use a calculator to simplify a rational expression?

A: Yes, you can use a calculator to simplify a rational expression. However, it is always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Q: How do I check my work when simplifying a rational expression?

A: To check your work when simplifying a rational expression, you can plug in a value for the variable and see if the expression simplifies to the correct value. You can also use a calculator to check your work.

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

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

Canceling out common factors without considering the entire expression.
Not having a common denominator when subtracting fractions.
Not factoring the numerator and denominator correctly.
Not combining the remaining terms correctly.

By following these tips and avoiding common mistakes, you can simplify rational expressions with confidence.