Write the Final Expression

Oct 13, 2025 by magicalceylontours.com 81 views

Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the rules of addition and subtraction. In this case, we are given two rational expressions that need to be added together. The expressions are:

$\frac{3x^2 - 2x - 1}{x-5}$ and $\frac{-2x^2 + x - 5}{x-5}$

Key Concepts

To add these expressions, we need to follow the rules of adding fractions. The key concept here is that the denominators of both expressions are the same, which is $(x-5)$ . This means that we can directly add the numerators and keep the denominator the same.

Step-by-Step Solution

To find the sum of the given expressions, we need to follow these steps:

Add the Numerators: The first step is to add the numerators of both expressions. The numerator of the first expression is $3x^2 - 2x - 1$ , and the numerator of the second expression is $-2x^2 + x - 5$ .

(3x^2 - 2x - 1) + (-2x^2 + x - 5)

Combine Like Terms: Now, we need to combine like terms in the numerator. The like terms are the terms with the same variable and exponent.

(3x^2 - 2x^2) + (-2x + x) + (-1 - 5)

Simplify the Numerator: After combining like terms, we get:

x^2 - x - 6

Write the Final Expression: Now that we have the simplified numerator, we can write the final expression by keeping the denominator the same.

$\frac{x^2 - x - 6}{x-5}$

Conclusion

In conclusion, the sum of the given algebraic expressions is $\frac{x^2 - x - 6}{x-5}$ . This is the final answer.

Answer Choice

The correct answer is:

D. $\frac{x^2 - x - 6}{x-5}$

Why is this the Correct Answer?

This is the correct answer because we followed the rules of adding fractions and simplified the numerator correctly. The denominator remains the same, which is $(x-5)$ .

What if the Denominators were Different?

If the denominators were different, we would need to find a common denominator before adding the expressions. However, in this case, the denominators are the same, which makes it easier to add the expressions.

Real-World Applications

This concept is essential in real-world applications, such as finance and engineering. For example, in finance, we may need to add two or more investment returns to find the total return. In engineering, we may need to add two or more forces to find the total force.

Common Mistakes

One common mistake is to forget to simplify the numerator after combining like terms. Another mistake is to forget to keep the denominator the same.

Tips and Tricks

To avoid common mistakes, make sure to simplify the numerator correctly and keep the denominator the same. Also, make sure to follow the rules of adding fractions.

Practice Problems

Here are some practice problems to help you understand the concept better:

Add the following expressions: $\frac{2x^2 + 3x - 1}{x+2}$ and $\frac{x^2 - 2x + 3}{x+2}$
Add the following expressions: $\frac{3x^2 - 2x - 1}{x-1}$ and $\frac{-2x^2 + x - 5}{x-1}$

Conclusion

In conclusion, adding algebraic expressions is a fundamental concept in mathematics. By following the rules of adding fractions and simplifying the numerator correctly, we can find the sum of two or more expressions. This concept is essential in real-world applications and can be practiced with the help of practice problems.

Q: What are algebraic expressions?

A: Algebraic expressions are mathematical expressions that contain variables and constants. They can be added, subtracted, multiplied, and divided, just like numbers.

Q: What is the rule for adding algebraic expressions?

A: The rule for adding algebraic expressions is to add the numerators and keep the denominator the same. If the denominators are different, we need to find a common denominator before adding the expressions.

Q: How do I add algebraic expressions with different denominators?

A: To add algebraic expressions with different denominators, we need to find a common denominator. This can be done by multiplying the denominators together. For example, if we have $\frac{2x^2 + 3x - 1}{x+2}$ and $\frac{x^2 - 2x + 3}{x-1}$ , we need to find a common denominator, which is $(x+2)(x-1)$ .

Q: How do I simplify the numerator after adding algebraic expressions?

A: To simplify the numerator after adding algebraic expressions, we need to combine like terms. Like terms are terms with the same variable and exponent. For example, if we have $(2x^2 + 3x - 1) + (x^2 - 2x + 3)$ , we can combine like terms to get $3x^2 + x - 4$ .

Q: What is the difference between adding and subtracting algebraic expressions?

A: The difference between adding and subtracting algebraic expressions is that when we subtract, we need to change the sign of the second expression. For example, if we have $\frac{2x^2 + 3x - 1}{x+2} - \frac{x^2 - 2x + 3}{x+2}$ , we need to change the sign of the second expression to get $\frac{2x^2 + 3x - 1}{x+2} + (-\frac{x^2 - 2x + 3}{x+2})$ .

Q: Can I add algebraic expressions with variables in the denominator?

A: Yes, you can add algebraic expressions with variables in the denominator. However, you need to be careful when simplifying the numerator. For example, if we have $\frac{2x^2 + 3x - 1}{x+2} + \frac{x^2 - 2x + 3}{x+2}$ , we need to simplify the numerator carefully to avoid getting a fraction with a variable in the denominator.

Q: How do I check my answer when adding algebraic expressions?

A: To check your answer when adding algebraic expressions, you can plug in a value for the variable and simplify the expression. For example, if we have $\frac{2x^2 + 3x - 1}{x+2} + \frac{x^2 - 2x + 3}{x+2}$ , we can plug in $x=1$ and simplify the expression to get $\frac{4 + 3 - 1}{3} = \frac{6}{3} = 2$ .

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

A: Some common mistakes to avoid when adding algebraic expressions include:

Forgetting to simplify the numerator after combining like terms
Forgetting to keep the denominator the same
Not changing the sign of the second expression when subtracting
Not being careful when simplifying the numerator with variables in the denominator

Q: How can I practice adding algebraic expressions?

A: You can practice adding algebraic expressions by working through examples and exercises in your textbook or online resources. You can also try creating your own problems and solving them to practice your skills.

Q: What are some real-world applications of adding algebraic expressions?

A: Some real-world applications of adding algebraic expressions include:

Finance: Adding investment returns to find the total return
Engineering: Adding forces to find the total force
Science: Adding measurements to find the total measurement

Q: Can I use a calculator to add algebraic expressions?

A: Yes, you can use a calculator to add algebraic expressions. However, it's always a good idea to check your answer by plugging in a value for the variable and simplifying the expression.

Q: How can I use technology to help me add algebraic expressions?

A: You can use technology such as graphing calculators or computer algebra systems to help you add algebraic expressions. These tools can simplify the expression and give you the answer quickly and accurately.