Not rewriting the expression as a product of two fractions

Oct 13, 2025 by talynt.org 94 views

$Not rewriting the expression as a product of two fractions$

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Introduction

When dealing with complex rational expressions, it's essential to simplify them to make them more manageable and easier to work with. In this article, we'll focus on simplifying the expression $\frac{x+3}{x^2-2x-3} \div \frac{x^2+2x-3}{x+1}$ , given that no denominator equals zero.

Understanding the Problem

The given expression involves the division of two rational expressions. To simplify this expression, we'll use the rule for dividing fractions, which states that dividing by a fraction is the same as multiplying by its reciprocal.

Step 1: Rewrite the Expression

The first step is to rewrite the expression as a product of two fractions. We'll do this by inverting the second fraction and changing the division sign to a multiplication sign.

\frac{x+3}{x^2-2x-3} \div \frac{x^2+2x-3}{x+1} = \frac{x+3}{x^2-2x-3} \cdot \frac{x+1}{x^2+2x-3}

Step 2: Multiply the Numerators and Denominators

Now that we have the expression as a product of two fractions, we can multiply the numerators and denominators separately.

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

Step 3: Simplify the Numerator

To simplify the numerator, we'll multiply the two binomials using the FOIL method.

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

Step 4: Simplify the Denominator

To simplify the denominator, we'll multiply the two quadratic expressions.

(x^2-2x-3)(x^2+2x-3) = x^4 + 2x^3 - 3x^2 - 2x^3 - 4x^2 + 6x - 3x^2 - 6x + 9

= x^4 - 9x^2 + 9

Step 5: Simplify the Expression

Now that we have the simplified numerator and denominator, we can rewrite the expression.

\frac{x^2 + 4x + 3}{x^4 - 9x^2 + 9}

Step 6: Factor the Numerator and Denominator

To simplify the expression further, we can factor the numerator and denominator.

\frac{(x+1)(x+3)}{(x^2-9)(x^2+1)}

Step 7: Simplify the Expression

Now that we have the factored numerator and denominator, we can simplify the expression.

\frac{(x+1)(x+3)}{(x^2-9)(x^2+1)} = \frac{(x+1)(x+3)}{(x-3)(x+3)(x^2+1)}

Step 8: Cancel Common Factors

We can cancel the common factor of $(x+3)$ from the numerator and denominator.

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

Conclusion

In conclusion, the simplified expression is $\frac{(x+1)}{(x-3)(x^2+1)}$ . This expression is equivalent to the original expression $\frac{x+3}{x^2-2x-3} \div \frac{x^2+2x-3}{x+1}$ , given that no denominator equals zero.

Final Answer

The final answer is $\boxed{\frac{(x+1)}{(x-3)(x^2+1)}}$ .

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Introduction

In our previous article, we explored the process of simplifying complex rational expressions. We walked through a step-by-step guide on how to simplify the expression $\frac{x+3}{x^2-2x-3} \div \frac{x^2+2x-3}{x+1}$ , given that no denominator equals zero. In this article, we'll answer some frequently asked questions related to simplifying complex rational expressions.

Q&A

Q: What is the first step in simplifying a complex rational expression?

A: The first step in simplifying a complex rational expression is to rewrite the expression as a product of two fractions. This involves inverting the second fraction and changing the division sign to a multiplication sign.

Q: How do I multiply the numerators and denominators of two fractions?

A: To multiply the numerators and denominators of two fractions, you simply multiply the numerators together and the denominators together. For example, if you have the expression $\frac{a}{b} \cdot \frac{c}{d}$ , you would multiply the numerators to get $ac$ and the denominators to get $bd$ .

Q: What is the FOIL method, and how do I use it to multiply binomials?

A: The FOIL method is a technique for multiplying two binomials. It stands for "First, Outer, Inner, Last," and it involves multiplying the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms. For example, if you have the expression $(x+3)(x+1)$ , you would use the FOIL method to get $x^2 + x + 3x + 3$ .

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you can try factoring it. If the expression can be factored, you can rewrite it as a product of two binomials. For example, if you have the expression $x^2 + 4x + 3$ , you can factor it as $(x+1)(x+3)$ .

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

A: A rational expression is a fraction that contains variables in the numerator or denominator. A complex rational expression is a rational expression that contains multiple fractions or other complex expressions.

Q: How do I cancel common factors in a rational expression?

A: To cancel common factors in a rational expression, you need to identify the common factors in the numerator and denominator. You can then cancel these factors by dividing them out of the expression. For example, if you have the expression $\frac{(x+1)(x+3)}{(x-3)(x+3)(x^2+1)}$ , you can cancel the common factor of $(x+3)$ to get $\frac{(x+1)}{(x-3)(x^2+1)}$ .

Common Mistakes to Avoid

When simplifying complex rational expressions, there are several common mistakes to avoid. Here are a few:

Not rewriting the expression as a product of two fractions: This is the first step in simplifying a complex rational expression. If you don't rewrite the expression as a product of two fractions, you may end up with a complicated expression that is difficult to simplify.
Not multiplying the numerators and denominators correctly: When multiplying the numerators and denominators of two fractions, make sure to multiply the numerators together and the denominators together.
Not factoring the numerator and denominator: Factoring the numerator and denominator can help you simplify the expression and cancel common factors.
Not canceling common factors: Canceling common factors is an important step in simplifying a rational expression. Make sure to identify the common factors in the numerator and denominator and cancel them out.

Conclusion

Simplifying complex rational expressions can be a challenging task, but with practice and patience, you can master the techniques and strategies involved. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex rational expressions with ease.

Final Tips

Here are a few final tips to keep in mind when simplifying complex rational expressions:

Take your time: Simplifying complex rational expressions can be a time-consuming process. Don't rush through it, and make sure to double-check your work.
Use a step-by-step approach: Break down the expression into smaller parts and simplify each part separately.
Use factoring to simplify the expression: Factoring the numerator and denominator can help you simplify the expression and cancel common factors.
Check your work: Once you've simplified the expression, make sure to check your work by plugging in values for the variables.

By following these tips and avoiding common mistakes, you can simplify complex rational expressions with ease and confidence.