Subtract: $\frac{3z-1}{z+4} - \frac{2}{z-4}$

Feb 27, 2025 by ADMIN 45 views

$Subtract: $\frac{3z-1}{z+4} - \frac{2}{z-4}$$

Introduction

When dealing with algebraic expressions, particularly those involving fractions, it's essential to understand the rules for subtracting them. In this article, we will delve into the process of subtracting two fractions with different denominators, focusing on the given expression: $\frac{3z-1}{z+4} - \frac{2}{z-4}$ . We will break down the steps involved in subtracting these fractions and provide a clear explanation of the process.

Understanding the Problem

To subtract the given fractions, we need to first find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators, which in this case are $z+4$ and $z-4$ . To find the LCM, we can list the multiples of each denominator and identify the smallest multiple that appears in both lists.

Finding the Common Denominator

The multiples of $z+4$ are: $z+4, 2(z+4), 3(z+4), \ldots$

The multiples of $z-4$ are: $z-4, 2(z-4), 3(z-4), \ldots$

By examining these lists, we can see that the smallest multiple that appears in both lists is $2(z^2+4z-4)$ , which is the LCM of $z+4$ and $z-4$ .

Rewriting the Fractions with the Common Denominator

Now that we have found the common denominator, we can rewrite each fraction with this denominator.

$\frac{3z-1}{z+4} = \frac{(3z-1)(z-4)}{(z+4)(z-4)} = \frac{3z^2-13z+4}{z^2-16}$

$\frac{2}{z-4} = \frac{2(z+4)}{(z-4)(z+4)} = \frac{2z+8}{z^2-16}$

Subtracting the Fractions

Now that we have rewritten the fractions with the common denominator, we can subtract them.

$\frac{3z-1}{z+4} - \frac{2}{z-4} = \frac{3z^2-13z+4}{z^2-16} - \frac{2z+8}{z^2-16}$

To subtract these fractions, we can subtract the numerators while keeping the denominator the same.

$\frac{3z^2-13z+4}{z^2-16} - \frac{2z+8}{z^2-16} = \frac{(3z^2-13z+4)-(2z+8)}{z^2-16}$

Simplifying the Expression

Now that we have subtracted the fractions, we can simplify the resulting expression.

$\frac{(3z^2-13z+4)-(2z+8)}{z^2-16} = \frac{3z^2-15z-4}{z^2-16}$

Conclusion

In this article, we have walked through the process of subtracting two fractions with different denominators. We found the common denominator, rewrote the fractions with this denominator, and then subtracted the fractions. Finally, we simplified the resulting expression. This process is essential in algebra and is used frequently in solving equations and manipulating expressions.

Common Mistakes to Avoid

When subtracting fractions, it's essential to find the common denominator and rewrite the fractions with this denominator. If you don't find the common denominator, you may end up with an incorrect result. Additionally, be careful when subtracting the numerators, as this can lead to errors.

Real-World Applications

Subtracting fractions is a fundamental skill that is used in a variety of real-world applications. For example, in physics, you may need to subtract fractions when calculating the velocity of an object. In engineering, you may need to subtract fractions when designing a system. In finance, you may need to subtract fractions when calculating interest rates.

Final Thoughts

Subtracting fractions is a critical skill that is used in a variety of mathematical and real-world applications. By following the steps outlined in this article, you can ensure that you are subtracting fractions correctly and accurately. Remember to find the common denominator, rewrite the fractions with this denominator, and then subtract the fractions. With practice and patience, you will become proficient in subtracting fractions and be able to apply this skill in a variety of contexts.

Introduction

In our previous article, we walked through the process of subtracting two fractions with different denominators. We found the common denominator, rewrote the fractions with this denominator, and then subtracted the fractions. In this article, we will answer some common questions that students may have when subtracting fractions.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the two denominators. To find the LCM, you can list the multiples of each denominator and identify the smallest multiple that appears in both lists.

Q: How do I find the common denominator?

A: To find the common denominator, you can use the following steps:

List the multiples of each denominator.
Identify the smallest multiple that appears in both lists.
This multiple is the common denominator.

Q: What if the denominators are not factorable?

A: If the denominators are not factorable, you can use the following steps to find the common denominator:

Multiply the denominators together.
Find the prime factorization of the product.
Identify the smallest multiple that appears in both lists.
This multiple is the common denominator.

Q: Can I subtract fractions with unlike denominators?

A: Yes, you can subtract fractions with unlike denominators. To do this, you need to find the common denominator and rewrite the fractions with this denominator.

Q: What if the fractions have different signs?

A: If the fractions have different signs, you need to change the sign of one of the fractions before subtracting. For example, if you have $\frac{3}{4} - \frac{2}{4}$ , you need to change the sign of the second fraction to get $\frac{3}{4} + \frac{2}{4}$ .

Q: Can I subtract fractions with variables in the denominators?

A: Yes, you can subtract fractions with variables in the denominators. To do this, you need to find the common denominator and rewrite the fractions with this denominator.

Q: What if the denominators are not polynomials?

A: If the denominators are not polynomials, you need to use a different method to find the common denominator. For example, if the denominators are rational expressions, you can use the following steps:

Multiply the denominators together.
Find the prime factorization of the product.
Identify the smallest multiple that appears in both lists.
This multiple is the common denominator.

Q: Can I subtract fractions with complex denominators?

A: Yes, you can subtract fractions with complex denominators. To do this, you need to find the common denominator and rewrite the fractions with this denominator.

Q: What if I make a mistake when subtracting fractions?

A: If you make a mistake when subtracting fractions, you can try the following steps to correct it:

Go back to the previous step and recheck your work.
Make sure you found the correct common denominator.
Make sure you rewrote the fractions with the correct denominator.
Make sure you subtracted the fractions correctly.

Conclusion

Common Mistakes to Avoid

When subtracting fractions, it's essential to avoid the following common mistakes:

Not finding the common denominator
Rewriting the fractions with the wrong denominator
Subtracting the fractions incorrectly
Not checking your work