Simplify The Expression:$\frac{b+x}{b^2+bx+x^2} - \frac{2x^3}{b^4+b^2x^2+x^4}$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. Rational expressions are a type of algebraic expression that contains variables in the numerator and denominator. In this article, we will focus on simplifying a specific rational expression involving two fractions. We will break down the problem into manageable steps and use various techniques to simplify the expression.

The Given Expression

The given expression is:

b+xb2+bx+x2βˆ’2x3b4+b2x2+x4\frac{b+x}{b^2+bx+x^2} - \frac{2x^3}{b^4+b^2x^2+x^4}

Our goal is to simplify this expression by combining the two fractions.

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators of both fractions. Let's start with the first fraction:

b2+bx+x2b^2+bx+x^2

We can factor this quadratic expression as:

(b+x)(b+x)(b+x)(b+x)

So, the first fraction becomes:

b+x(b+x)(b+x)\frac{b+x}{(b+x)(b+x)}

Now, let's factor the denominator of the second fraction:

b4+b2x2+x4b^4+b^2x^2+x^4

We can factor this quartic expression as:

(b2+x2)(b2+x2)(b^2+x^2)(b^2+x^2)

However, we can further factor it as:

((b+x)2)((bβˆ’x)2)((b+x)^2)((b-x)^2)

But since we are looking for a factorization that can be used to cancel out common factors, we will use the first factorization:

(b2+x2)(b2+x2)(b^2+x^2)(b^2+x^2)

So, the second fraction becomes:

2x3(b2+x2)(b2+x2)\frac{2x^3}{(b^2+x^2)(b^2+x^2)}

Step 2: Find Common Factors

Now that we have factored the denominators, let's find common factors between the two fractions. We can see that both fractions have a common factor of (b+x)(b+x) in the numerator and (b2+x2)(b^2+x^2) in the denominator.

Step 3: Cancel Out Common Factors

We can cancel out the common factors between the two fractions:

b+x(b+x)(b+x)βˆ’2x3(b2+x2)(b2+x2)\frac{b+x}{(b+x)(b+x)} - \frac{2x^3}{(b^2+x^2)(b^2+x^2)}

1b+xβˆ’2x3(b2+x2)2\frac{1}{b+x} - \frac{2x^3}{(b^2+x^2)^2}

Step 4: Find a Common Denominator

To combine the two fractions, we need to find a common denominator. The least common multiple (LCM) of (b+x)(b+x) and (b2+x2)2(b^2+x^2)^2 is (b+x)(b2+x2)2(b+x)(b^2+x^2)^2.

Step 5: Rewrite the Fractions with a Common Denominator

We can rewrite the fractions with a common denominator:

1b+xβˆ’2x3(b2+x2)2\frac{1}{b+x} - \frac{2x^3}{(b^2+x^2)^2}

(b2+x2)2(b+x)(b2+x2)2βˆ’2x3(b+x)(b+x)(b2+x2)2\frac{(b^2+x^2)^2}{(b+x)(b^2+x^2)^2} - \frac{2x^3(b+x)}{(b+x)(b^2+x^2)^2}

Step 6: Combine the Fractions

Now that we have a common denominator, we can combine the fractions:

(b2+x2)2βˆ’2x3(b+x)(b+x)(b2+x2)2\frac{(b^2+x^2)^2 - 2x^3(b+x)}{(b+x)(b^2+x^2)^2}

Step 7: Simplify the Numerator

We can simplify the numerator by expanding and combining like terms:

(b2+x2)2βˆ’2x3(b+x)(b^2+x^2)^2 - 2x^3(b+x)

=(b4+2b2x2+x4)βˆ’2bx3βˆ’2x4= (b^4 + 2b^2x^2 + x^4) - 2bx^3 - 2x^4

=b4+2b2x2+x4βˆ’2bx3βˆ’2x4= b^4 + 2b^2x^2 + x^4 - 2bx^3 - 2x^4

=b4βˆ’2bx3+2b2x2βˆ’x4= b^4 - 2bx^3 + 2b^2x^2 - x^4

Step 8: Factor the Numerator

We can factor the numerator as:

b4βˆ’2bx3+2b2x2βˆ’x4b^4 - 2bx^3 + 2b^2x^2 - x^4

=(b2βˆ’x2)2= (b^2 - x^2)^2

Step 9: Simplify the Expression

Now that we have simplified the numerator, we can simplify the expression:

(b2+x2)2βˆ’2x3(b+x)(b+x)(b2+x2)2\frac{(b^2+x^2)^2 - 2x^3(b+x)}{(b+x)(b^2+x^2)^2}

=(b2βˆ’x2)2(b+x)(b2+x2)2= \frac{(b^2 - x^2)^2}{(b+x)(b^2+x^2)^2}

Conclusion

In this article, we simplified a rational expression involving two fractions. We broke down the problem into manageable steps and used various techniques to simplify the expression. We factored the denominators, found common factors, canceled out common factors, found a common denominator, and combined the fractions. Finally, we simplified the numerator and factored it to obtain the final simplified expression.

Final Answer

The final simplified expression is:

(b2βˆ’x2)2(b+x)(b2+x2)2\frac{(b^2 - x^2)^2}{(b+x)(b^2+x^2)^2}

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Introduction

In our previous article, we simplified a rational expression involving two fractions. We broke down the problem into manageable steps and used various techniques to simplify the expression. In this article, we will answer some common questions related to simplifying rational expressions.

Q&A

Q: What is a rational expression?

A: A rational expression is an algebraic expression that contains variables in the numerator and denominator. It is a fraction of two polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

  1. Factor the denominators.
  2. Find common factors between the two fractions.
  3. Cancel out common factors.
  4. Find a common denominator.
  5. Rewrite the fractions with a common denominator.
  6. Combine the fractions.
  7. Simplify the numerator.
  8. Factor the numerator.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression, on the other hand, is an algebraic expression that contains variables in the numerator and denominator.

Q: How do I know if a rational expression is simplified?

A: A rational expression is simplified when the numerator and denominator have no common factors other than 1.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful when canceling out common factors.

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

A: The LCM of two expressions is the smallest expression that is a multiple of both expressions.

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

A: To find the LCM of two expressions, you need to list the multiples of each expression and find the smallest common multiple.

Q: Can I simplify a rational expression with a negative exponent?

A: Yes, you can simplify a rational expression with a negative exponent. However, you need to be careful when canceling out common factors.

Q: What is the difference between a rational expression and an algebraic expression?

A: An algebraic expression is a general term that refers to any expression that contains variables and constants. A rational expression, on the other hand, is a specific type of algebraic expression that contains variables in the numerator and denominator.

Example Problems

Problem 1

Simplify the rational expression:

x2+3x+2x2+4x+3\frac{x^2 + 3x + 2}{x^2 + 4x + 3}

Solution

To simplify the rational expression, we need to factor the denominators:

x2+4x+3=(x+1)(x+3)x^2 + 4x + 3 = (x+1)(x+3)

We can then cancel out common factors:

x2+3x+2(x+1)(x+3)=x+2x+3\frac{x^2 + 3x + 2}{(x+1)(x+3)} = \frac{x+2}{x+3}

Problem 2

Simplify the rational expression:

2x2+5x+3x2+2x+1\frac{2x^2 + 5x + 3}{x^2 + 2x + 1}

Solution

To simplify the rational expression, we need to factor the denominators:

x2+2x+1=(x+1)2x^2 + 2x + 1 = (x+1)^2

We can then cancel out common factors:

2x2+5x+3(x+1)2=2x+3x+1\frac{2x^2 + 5x + 3}{(x+1)^2} = \frac{2x+3}{x+1}

Conclusion

In this article, we answered some common questions related to simplifying rational expressions. We discussed the difference between a rational expression and a rational number, and provided examples of how to simplify rational expressions. We also discussed the importance of finding a common denominator and canceling out common factors.

Final Answer

The final answer is that simplifying rational expressions is an important skill in algebra that can be used to solve equations and inequalities more efficiently. By following the steps outlined in this article, you can simplify rational expressions and solve problems more easily.