**Title: Understanding Equivalent Expressions****Objective: For non-zero denominators, determine which expression is equivalent to $ \frac{1}{a} - \frac{2a}{a+1} $.**The problem provides a mathematical expression and asks to identify which option among the provided choices is equivalent to the given expression. The expression in question is:\[ \frac{1}{a} - \frac{2a}{a+1} \]### Options1. \[ \frac{-2a^3 + 1}{a(a + 1)} \]2. \[ 1 - \frac{2a}{1} \]3. \[ \frac{1 - 2a}{a(a + 1)} \]4. \[ \frac{-2a^2 + a + 1}{a(a + 1)} \]The task is to verify which of the four options correctly simplifies to the given expression considering non-zero denominators.### Detailed Analysis of Each Option- **Option 1:** $\frac{-2a^3 + 1}{a(a + 1)}$ Analyze the numerator and denominator components to see if it matches the given algebraic expression when simplified. - **Option 2:** $1 - \frac{2a}{1}$ This option simplifies straightforwardly to $1 - 2a$. Compare this with the given expression to verify its equivalence. - **Option 3:** $\frac{1 - 2a}{a(a + 1)}$ Here, assess both the numerator $(1 - 2a)$ and the denominator $(a(a + 1))$, and compare it against the target expression. - **Option 4:** $\frac{-2a^2 + a + 1}{a(a + 1)}$ Break down the numerator $(-2a^2 + a + 1)$ and match it with both components of the target expression when combined over the common denominator.By determining the steps each option undergoes to see if they simplify down to $\frac{1}{a} - \frac{2a}{a+1}$, we'll identify the equivalent expression.### ConclusionThis exercise tests your ability to algebraically manipulate expressions, identify common denominators, and correctly simplify to determine equivalency. Through this

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Description: **Title: Understanding Equivalent Expressions****Objective: For non-zero denominators, determine which expression is equivalent to $ \frac{1}{a} - \frac{2a}{a+1} $.**The problem provides a mathematical expression and asks to identify which option

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Answered: For non-zero denominators, which of the…

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For non-zero denominators, which of the following is equivalent to 1 2a a a + 1 -2a3 +1 a(a + 1) 1- 2a 1- 2a ala + 1). -2a2+ a +1 a(a + 1)

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

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Question

**Title: Understanding Equivalent Expressions**

**Objective: For non-zero denominators, determine which expression is equivalent to $ \frac{1}{a} - \frac{2a}{a+1} $.**

The problem provides a mathematical expression and asks to identify which option among the provided choices is equivalent to the given expression. The expression in question is:

\[ \frac{1}{a} - \frac{2a}{a+1} \]

### Options

1. \[ \frac{-2a^3 + 1}{a(a + 1)} \]
2. \[ 1 - \frac{2a}{1} \]
3. \[ \frac{1 - 2a}{a(a + 1)} \]
4. \[ \frac{-2a^2 + a + 1}{a(a + 1)} \]

The task is to verify which of the four options correctly simplifies to the given expression considering non-zero denominators.

### Detailed Analysis of Each Option

- **Option 1:** $\frac{-2a^3 + 1}{a(a + 1)}$

Analyze the numerator and denominator components to see if it matches the given algebraic expression when simplified.

- **Option 2:** $1 - \frac{2a}{1}$

This option simplifies straightforwardly to $1 - 2a$. Compare this with the given expression to verify its equivalence.

- **Option 3:** $\frac{1 - 2a}{a(a + 1)}$

Here, assess both the numerator $(1 - 2a)$ and the denominator $(a(a + 1))$, and compare it against the target expression.

- **Option 4:** $\frac{-2a^2 + a + 1}{a(a + 1)}$

Break down the numerator $(-2a^2 + a + 1)$ and match it with both components of the target expression when combined over the common denominator.

By determining the steps each option undergoes to see if they simplify down to $\frac{1}{a} - \frac{2a}{a+1}$, we'll identify the equivalent expression.

### Conclusion

This exercise tests your ability to algebraically manipulate expressions, identify common denominators, and correctly simplify to determine equivalency. Through this

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