Simplify (a X^2+b)/(2 X-1)+(a X^2-b)/(2 X+1)+(4 A X^2)/(1-4 X^2) Expression A Step By Step Guide

Hey guys! Ever stumbled upon a math problem that looks like a tangled mess of fractions and variables? Today, we're going to untangle one such expression and simplify it step by step. The problem we're tackling is this real doozy:

$$\frac{a x^2+b}{2 x-1}+\frac{a x^2-b}{2 x+1}+\frac{4 a x^2}{1-4 x^2}$$

Don't worry, it might look intimidating, but with a little algebraic magic, we can make it much simpler. So, grab your pencils, and let's dive in!

1. Understanding the Expression

Before we start manipulating anything, let's take a good look at the expression. We have three fractions, each with its own numerator and denominator. The denominators are 2x - 1, 2x + 1, and 1 - 4x^2. Notice anything interesting about that last one? It looks suspiciously like a difference of squares! Spotting these patterns is key to simplifying complex expressions.

The given expression involves adding three rational expressions. To effectively simplify such expressions, it's crucial to identify common denominators and combine the fractions. Our primary goal is to find a common denominator for all three fractions, which will allow us to add them together. This involves a bit of algebraic manipulation and recognizing key patterns. Let's break down each component of the expression:

1. The first term is ${\frac{a x^2+b}{2 x-1}}$. This fraction has a quadratic expression in the numerator and a linear expression in the denominator.
2. The second term is ${\frac{a x^2-b}{2 x+1}}$. Similar to the first term, this also has a quadratic numerator and a linear denominator, but with a slightly different structure.
3. The third term is ${\frac{4 a x^2}{1-4 x^2}}$. This fraction is particularly interesting because the denominator ${1 - 4x^2}$ can be recognized as a difference of squares.

Recognizing the difference of squares is a crucial step in simplifying this expression. The expression ${1 - 4x^2}$ can be factored into ${(1 - 2x)(1 + 2x)}$ or ${(-1)(2x - 1)(2x + 1)}$. This factorization will be essential in finding a common denominator. Combining rational expressions often involves finding the least common multiple (LCM) of the denominators. In this case, by factoring ${1 - 4x^2}$, we can see how it relates to the other denominators, making it easier to find the LCM. The ultimate aim is to add these fractions together, which requires them to have the same denominator. Once we achieve a common denominator, we can combine the numerators and simplify the resulting expression. This often involves expanding terms, combining like terms, and possibly further factoring to reach the simplest form. Simplifying this expression is not just an exercise in algebra; it is a process that highlights the importance of recognizing patterns, strategic manipulation, and attention to detail. By carefully working through each step, we can transform a seemingly complex expression into something much more manageable and elegant. Understanding the structure and relationships between the terms is the cornerstone of effective simplification in algebra.

2. Finding a Common Denominator

The key to adding fractions is, you guessed it, a common denominator! Looking at our denominators (2x - 1), (2x + 1), and (1 - 4x^2), we can see that 1 - 4x^2 is actually (1 - 2x)(1 + 2x). To make things even clearer, we can factor out a -1 to get -(2x - 1)(2x + 1). This is super helpful because it shows us that the common denominator we need is (2x - 1)(2x + 1).

To find the common denominator, we first recognize that ${1 - 4x^2}$ can be factored as a difference of squares: ${1 - 4x^2 = (1 - 2x)(1 + 2x)}$. We also observe that ${(1 - 2x)}$ is the negative of ${(2x - 1)}$, meaning ${(1 - 2x) = -(2x - 1)}$. Thus, we can rewrite the factored form as ${1 - 4x^2 = -(2x - 1)(2x + 1)}$. This insight is crucial because it allows us to see the relationship between the third denominator and the first two. The common denominator will therefore be ${(2x - 1)(2x + 1)}$. Now, we need to rewrite each fraction with this common denominator. For the first fraction, ${\frac{a x^2+b}{2 x-1}}$, we multiply both the numerator and the denominator by ${(2x + 1)}$:

$$\frac{a x^2+b}{2 x-1} \cdot \frac{2x + 1}{2x + 1} = \frac{(a x^2+b)(2x + 1)}{(2x - 1)(2x + 1)}$$

For the second fraction, ${\frac{a x^2-b}{2 x+1}}$, we multiply both the numerator and the denominator by ${(2x - 1)}$:

$$\frac{a x^2-b}{2 x+1} \cdot \frac{2x - 1}{2x - 1} = \frac{(a x^2-b)(2x - 1)}{(2x - 1)(2x + 1)}$$

For the third fraction, ${\frac{4 a x^2}{1-4 x^2}}$, we rewrite the denominator as ${-(2x - 1)(2x + 1)}$. To get the common denominator, we multiply both the numerator and the denominator by ${-1}$:

$$\frac{4 a x^2}{1-4 x^2} = \frac{4 a x^2}{-(2x - 1)(2x + 1)} = \frac{-4 a x^2}{(2x - 1)(2x + 1)}$$

By doing this, all three fractions now have the same denominator, ${(2x - 1)(2x + 1)}$, which sets us up to combine the numerators. Finding a common denominator is a fundamental technique in adding fractions, whether they involve simple numbers or complex algebraic expressions. The key is to identify the least common multiple (LCM) of the denominators, which allows us to rewrite each fraction with a common base. In our case, recognizing the difference of squares and strategically using negative signs allowed us to align the denominators efficiently. This step is crucial for the subsequent addition and simplification of the expression. By ensuring that all fractions share a common denominator, we can accurately combine their numerators and move closer to the simplified form of the original expression.

3. Rewriting the Fractions

Now we need to rewrite each fraction with the common denominator. This means multiplying the numerator and denominator of the first fraction by (2x + 1), the second fraction by (2x - 1), and the third fraction (carefully!) by -1 (since we factored out a -1 earlier). This gives us:

• First fraction: $\frac{(a x^2+b)(2x + 1)}{(2 x-1)(2 x+1)}$
• Second fraction: $\frac{(a x^2-b)(2x - 1)}{(2 x-1)(2 x+1)}$
• Third fraction: $\frac{-4 a x^2}{(2 x-1)(2 x+1)}$

Notice how the third fraction's sign changed because we multiplied by -1. This is a crucial step to ensure we're working with the correct signs when we combine the fractions. Rewriting the fractions with a common denominator is a critical step in the process of simplifying the given expression. This ensures that we can accurately add the numerators together, as the fractions now have a consistent base. Let's delve into the specifics of how each fraction is transformed:

1. First Fraction: ${\frac{a x^2+b}{2 x-1}}$

   To get the common denominator ${(2x - 1)(2x + 1)}$, we need to multiply both the numerator and the denominator by ${(2x + 1)}$. This gives us:

   $$\frac{(a x^2+b)(2x + 1)}{(2 x-1)(2 x+1)}$$

   The numerator now becomes a product of a quadratic expression and a linear expression, which we will expand later to combine like terms.

2. Second Fraction: ${\frac{a x^2-b}{2 x+1}}$

   Similarly, we multiply both the numerator and the denominator by ${(2x - 1)}$ to achieve the common denominator:

   $$\frac{(a x^2-b)(2x - 1)}{(2 x-1)(2 x+1)}$$

   Again, the numerator is a product of a quadratic and a linear expression, which will require expansion and simplification in the subsequent steps.

3. Third Fraction: ${\frac{4 a x^2}{1-4 x^2}}$

   Here, we recall that ${1 - 4x^2 = -(2x - 1)(2x + 1)}$. To align this with our desired common denominator ${(2x - 1)(2x + 1)}$, we multiply the numerator and the denominator by ${-1}$. This gives us:

   $$\frac{-4 a x^2}{(2 x-1)(2 x+1)}$$

   Multiplying by ${-1}$ is crucial here as it corrects the sign and ensures that we can combine this term correctly with the other two fractions. By rewriting each fraction in this way, we set the stage for the next step: adding the numerators. The common denominator acts as a foundation, allowing us to focus on the numerators and their algebraic manipulation. This process highlights the importance of careful multiplication and attention to signs to ensure the accuracy of the resulting expressions. Each rewritten fraction is now in a form that facilitates the final simplification, bringing us closer to the solution.

4. Combining the Numerators

Now for the fun part! Since all the fractions have the same denominator, we can combine the numerators. This means adding them all together:

$$(a x^2+b)(2x + 1) + (a x^2-b)(2x - 1) - 4 a x^2$$

We need to expand these products and then simplify by combining like terms. This is where careful algebra is key! Combining the numerators is the pivotal step where we bring all the individual fractions together into a single expression. With a common denominator in place, we can now add the numerators:

$$(a x^2+b)(2x + 1) + (a x^2-b)(2x - 1) - 4 a x^2$$

This expression represents the sum of the three numerators we derived in the previous step. The next critical task is to expand these products. Expanding involves multiplying out the terms within the parentheses. Let’s expand each product step by step:

1. Expanding ${(a x^2+b)(2x + 1)}$:

   We apply the distributive property (also known as the FOIL method) to multiply each term in the first expression by each term in the second expression:

   $$(a x^2)(2x) + (a x^2)(1) + (b)(2x) + (b)(1) = 2 a x^3 + a x^2 + 2 b x + b$$

2. Expanding ${(a x^2-b)(2x - 1)}$:

   Similarly, we multiply each term in the first expression by each term in the second expression:

   $$(a x^2)(2x) + (a x^2)(-1) + (-b)(2x) + (-b)(-1) = 2 a x^3 - a x^2 - 2 b x + b$$

3. The third term, ${-4 a x^2}$, remains as it is.

Now that we have expanded the products, we can rewrite the combined numerator expression:

$$2 a x^3 + a x^2 + 2 b x + b + 2 a x^3 - a x^2 - 2 b x + b - 4 a x^2$$

The next step is to simplify by combining like terms. This involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. It’s a meticulous process that requires careful attention to detail to ensure that all terms are correctly accounted for. This simplification will lead us to a more manageable expression, making it easier to see if further factoring or simplification is possible.

5. Expanding and Simplifying

Let's expand those products:

• (a x^2 + b)(2x + 1) = 2ax^3 + ax^2 + 2bx + b
• (a x^2 - b)(2x - 1) = 2ax^3 - ax^2 - 2bx + b

Now, let's put it all together and simplify:

2ax^3 + ax^2 + 2bx + b + 2ax^3 - ax^2 - 2bx + b - 4ax^2

Notice how several terms cancel out! The ax^2 and -ax^2 cancel, the 2bx and -2bx cancel. We're left with:

4ax^3 - 4ax^2 + 2b

This is much simpler, right? The process of expanding and simplifying the combined numerator is a crucial step in reducing the expression to its simplest form. After combining the numerators, we had the expression:

$$2 a x^3 + a x^2 + 2 b x + b + 2 a x^3 - a x^2 - 2 b x + b - 4 a x^2$$

The first part of this process is expanding the products. We have already done this in the previous step, where we meticulously applied the distributive property to multiply out the terms. Now, we focus on simplifying the expression. This involves combining like terms, which means identifying terms with the same variable and exponent and then adding or subtracting their coefficients. Let’s go through the terms systematically:

1. Terms with ${x^3}$:

   We have ${2 a x^3}$ and ${2 a x^3}$. Combining these gives us:

   $$2 a x^3 + 2 a x^3 = 4 a x^3$$

2. Terms with ${x^2}$:

   We have ${a x^2}$, ${-a x^2}$, and ${-4 a x^2}$. Combining these gives us:

   $$a x^2 - a x^2 - 4 a x^2 = -4 a x^2$$

3. Terms with ${x}$:

   We have ${2 b x}$ and ${-2 b x}$. These terms cancel each other out:

   $$2 b x - 2 b x = 0$$

4. Constant terms:

   We have ${b}$ and ${b}$. Combining these gives us:

   $$b + b = 2b$$

Putting it all together, we get the simplified numerator:

$$4 a x^3 - 4 a x^2 + 2b$$

This simplified numerator is significantly cleaner than the expanded form. By systematically combining like terms, we have reduced the complexity of the expression, making it easier to manage and analyze. This step is vital because it brings us closer to the final simplified form of the original expression. The process of simplifying not only makes the expression more concise but also reveals the underlying structure, which can be crucial for further analysis or application of the result.

6. Putting it All Together

Now we put the simplified numerator back over the common denominator:

$$\frac{4 a x^3 - 4 a x^2 + 2b}{(2 x-1)(2 x+1)}$$

Can we simplify further? Maybe! Let's factor a 2 out of the numerator:

$$\frac{2(2 a x^3 - 2 a x^2 + b)}{(2 x-1)(2 x+1)}$$

Unfortunately, it doesn't look like anything will cancel with the denominator. So, this is our simplified expression! Putting it all together means combining the simplified numerator with the common denominator to form the final expression. After expanding and simplifying the numerator, we arrived at:

$$4 a x^3 - 4 a x^2 + 2b$$

And we know the common denominator is:

$$(2x - 1)(2x + 1)$$

So, our combined expression is:

$$\frac{4 a x^3 - 4 a x^2 + 2b}{(2 x-1)(2 x+1)}$$

Now, we consider whether we can simplify further. Simplifying further often involves looking for common factors in the numerator and the denominator that can be canceled out. One common technique is to factor out any common numerical factors. In our numerator, we can see that each term has a factor of 2. So, we factor out a 2:

$$2(2 a x^3 - 2 a x^2 + b)$$

Now, the expression becomes:

$$\frac{2(2 a x^3 - 2 a x^2 + b)}{(2 x-1)(2 x+1)}$$

The next step is to see if any factors in the numerator can cancel with factors in the denominator. The denominator is a difference of squares, which expands to ${4x^2 - 1}$. The numerator is a cubic polynomial, and there’s no immediately obvious way to factor it further or see any common factors with the denominator. By examining the expression, we can see that the numerator and the denominator do not share any common factors that can be canceled out. This indicates that we have simplified the expression as much as possible. The final simplified form of the expression is therefore:

$$\frac{2(2 a x^3 - 2 a x^2 + b)}{(2 x-1)(2 x+1)}$$

This is our final answer, guys! Remember, simplifying complex expressions is all about breaking them down into smaller, manageable steps. By finding a common denominator, combining numerators, expanding, and simplifying, we were able to transform a complicated-looking expression into something much cleaner. Keep practicing, and you'll become a simplification master in no time!

7. Final Simplified Expression

Therefore, the simplified form of the given expression is:

$$\frac{2(2 a x^3 - 2 a x^2 + b)}{(2 x-1)(2 x+1)}$$

And that's it! We took a seemingly complex expression and simplified it step by step. Remember, guys, the key to success in algebra is to break down problems into smaller, more manageable steps. The final simplified expression is the result of a careful, step-by-step simplification process. After combining the fractions, expanding the products, and simplifying the numerator, we arrived at the expression:

$$\frac{2(2 a x^3 - 2 a x^2 + b)}{(2 x-1)(2 x+1)}$$

This expression represents the most reduced form of the original problem. Each step in the simplification was crucial to achieving this result. First, finding the common denominator ${(2x - 1)(2x + 1)}$ allowed us to combine the three fractions into one. This involved recognizing that ${1 - 4x^2}$ could be factored as a difference of squares, which simplified the process of finding the common denominator. Next, expanding the numerators required careful application of the distributive property. We meticulously multiplied each term in the binomials to obtain a polynomial expression. This step is prone to errors if not done carefully, so paying close attention to each multiplication is essential. Then, simplifying the numerator involved combining like terms. This process reduced the complexity of the numerator, making it easier to manage. We combined terms with the same powers of ${x}$, which resulted in a cleaner expression. Finally, after obtaining the simplified numerator, we checked for any common factors with the denominator. Factoring out the 2 from the numerator was a key step in seeing if further simplification was possible. However, after factoring out the 2, we observed that the resulting cubic polynomial did not share any common factors with the quadratic denominator. Therefore, the expression ${\frac{2(2 a x^3 - 2 a x^2 + b)}{(2 x-1)(2 x+1)}}$ is indeed the final simplified form. It represents the culmination of all our algebraic manipulations and provides a concise and manageable representation of the original, more complex expression. This process highlights the importance of methodical and careful algebra in simplifying mathematical expressions. By following these steps, we can transform complex problems into more understandable and solvable forms.