Perform The Operation And Combine To One Fraction. 3 X + 8 + 3 X X − 1 \frac{3}{x+8} + \frac{3x}{x-1} X + 8 3 ​ + X − 1 3 X ​ Submit Your Answer In The Box Below: □ \square □

Introduction

In mathematics, combining fractions is an essential skill that helps us simplify complex expressions and solve problems more efficiently. When we have multiple fractions with different denominators, we can combine them by finding a common denominator and adding or subtracting the numerators. In this article, we will focus on combining two fractions with different denominators, specifically $\frac{3}{x+8} + \frac{3x}{x-1}$.

Understanding the Problem

The given problem involves adding two fractions with different denominators: $\frac{3}{x+8}$ and $\frac{3x}{x-1}$. To combine these fractions, we need to find a common denominator, which is the least common multiple (LCM) of the two denominators. In this case, the LCM of $x+8$ and $x-1$ is $(x+8)(x-1)$.

Step 1: Find the Common Denominator

To find the common denominator, we need to multiply the two denominators together: $(x+8)(x-1)$. This will give us a new expression that we can use as the common denominator.

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

denom1 = x + 8
denom2 = x - 1

common_denom = denom1 * denom2
print(common_denom)

Step 2: Rewrite the Fractions with the Common Denominator

Now that we have the common denominator, we can rewrite each fraction with the new denominator. To do this, we need to multiply the numerator and denominator of each fraction by the necessary factors to get the common denominator.

# Rewrite the fractions with the common denominator
frac1 = (3 * (x - 1)) / common_denom
frac2 = (3x * (x + 8)) / common_denom
print(frac1)
print(frac2)

Step 3: Add the Fractions

Now that we have rewritten the fractions with the common denominator, we can add them together. To do this, we simply add the numerators and keep the common denominator.

# Add the fractions
result = (3 * (x - 1) + 3x * (x + 8)) / common_denom
print(result)

Simplifying the Result

The result we obtained in the previous step is a fraction with a common denominator. However, we can simplify this fraction further by combining like terms in the numerator.

# Simplify the result
simplified_result = sp.simplify(result)
print(simplified_result)

Conclusion

In this article, we learned how to combine two fractions with different denominators by finding a common denominator and adding or subtracting the numerators. We applied this concept to the problem $\frac{3}{x+8} + \frac{3x}{x-1}$ and obtained the simplified result $\frac{3x^2 + 21x - 24}{(x+8)(x-1)}$. This result can be further simplified by factoring the numerator and canceling out any common factors between the numerator and denominator.

Final Answer

Introduction

In our previous article, we explored the concept of combining fractions with different denominators. We learned how to find a common denominator, rewrite the fractions with the new denominator, and add them together. In this article, we will answer some frequently asked questions about combining fractions to help you better understand this concept.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that can be divided by both 4 and 6.

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

A: To find the LCM of two expressions, you can use the following steps:

1. Factor each expression into its prime factors.
2. Identify the common factors between the two expressions.
3. Multiply the common factors together to get the LCM.

For example, let's find the LCM of $x+8$ and $x-1$.

import sympy as sp

x = sp.symbols('x')

expr1 = x + 8
expr2 = x - 1

factored_expr1 = sp.factor(expr1)
factored_expr2 = sp.factor(expr2)

common_factors = sp.gcd(factored_expr1, factored_expr2)

lcm = common_factors * (x + 8) * (x - 1)
print(lcm)

Q: How do I rewrite a fraction with a different denominator?

A: To rewrite a fraction with a different denominator, you can multiply the numerator and denominator by the necessary factors to get the new denominator.

For example, let's rewrite the fraction $\frac{3}{x+8}$ with the denominator $x-1$.

# Define the variables
x = sp.symbols('x')

frac = 3 / (x + 8)

new_frac = (3 * (x - 1)) / ((x + 8) * (x - 1))
print(new_frac)

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you can follow these steps:

1. Find the LCM of the two denominators.
2. Rewrite each fraction with the LCM as the denominator.
3. Add the fractions together.

For example, let's add the fractions $\frac{3}{x+8}$ and $\frac{3x}{x-1}$.

# Define the variables
x = sp.symbols('x')

frac1 = 3 / (x + 8)
frac2 = 3x / (x - 1)

lcm = (x + 8) * (x - 1)

new_frac1 = (3 * (x - 1)) / lcm
new_frac2 = (3x * (x + 8)) / lcm

result = (new_frac1 + new_frac2)
print(result)

Q: Can I simplify a fraction after combining it with another fraction?

A: Yes, you can simplify a fraction after combining it with another fraction. To do this, you can factor the numerator and denominator, and then cancel out any common factors.

For example, let's simplify the fraction $\frac{3x^2 + 21x - 24}{(x+8)(x-1)}$.

# Define the variables
x = sp.symbols('x')

frac = (3x**2 + 21x - 24) / ((x + 8) * (x - 1))

factored_frac = sp.factor(frac)

simplified_frac = sp.cancel(factored_frac)
print(simplified_frac)

Conclusion

In this article, we answered some frequently asked questions about combining fractions to help you better understand this concept. We covered topics such as finding the least common multiple (LCM) of two expressions, rewriting fractions with different denominators, adding fractions with different denominators, and simplifying fractions after combining them with another fraction. By following these steps and practicing with examples, you can become more confident in your ability to combine fractions and solve problems involving fractions with different denominators.