Factorizing the Numerators

Oct 13, 2025 by miodeya.com 102 views

Introduction

Algebraic expressions can be complex and daunting, especially when dealing with fractions and variables. In this article, we will explore how to simplify a complex algebraic expression involving fractions and variables. We will use the given expression $\frac{5x^2+10x}{x^2+7x+12} \div \frac{x^2-4}{x^2+x-6}$ as an example and break it down into manageable steps.

Step 1: Factorize the Numerators and Denominators

To simplify the expression, we need to factorize the numerators and denominators of both fractions.

Factorizing the Numerators

The numerator of the first fraction is $5x^2+10x$ . We can factor out $5x$ from both terms:

5x^2+10x = 5x(x+2)

The numerator of the second fraction is $x^2-4$ . We can factor it as a difference of squares:

x^2-4 = (x-2)(x+2)

Factorizing the Denominators

The denominator of the first fraction is $x^2+7x+12$ . We can factor it as:

x^2+7x+12 = (x+3)(x+4)

The denominator of the second fraction is $x^2+x-6$ . We can factor it as:

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

Step 2: Rewrite the Expression with Factorized Numerators and Denominators

Now that we have factorized the numerators and denominators, we can rewrite the expression as:

\frac{5x(x+2)}{(x+3)(x+4)} \div \frac{(x-2)(x+2)}{(x-2)(x+3)}

Step 3: Simplify the Expression

To simplify the expression, we need to cancel out common factors between the numerators and denominators.

Canceling Out Common Factors

We can cancel out the $(x+2)$ term in the numerator of the first fraction with the $(x+2)$ term in the denominator of the second fraction:

\frac{5x(x+2)}{(x+3)(x+4)} \div \frac{(x-2)(x+2)}{(x-2)(x+3)} = \frac{5x}{(x+4)} \div \frac{(x-2)}{(x+3)}

We can also cancel out the $(x-2)$ term in the denominator of the second fraction with the $(x-2)$ term in the numerator of the second fraction:

\frac{5x}{(x+4)} \div \frac{(x-2)}{(x+3)} = \frac{5x}{(x+4)} \times \frac{(x+3)}{(x-2)}

Step 4: Multiply the Numerators and Denominators

To simplify the expression further, we need to multiply the numerators and denominators:

\frac{5x}{(x+4)} \times \frac{(x+3)}{(x-2)} = \frac{5x(x+3)}{(x+4)(x-2)}

Step 5: Simplify the Expression Further

We can simplify the expression further by canceling out common factors between the numerator and denominator:

\frac{5x(x+3)}{(x+4)(x-2)} = \frac{5x(x+3)}{(x+4)(x-2)} \times \frac{(x+4)}{(x+4)} \times \frac{(x-2)}{(x-2)}

This simplifies to:

\frac{5x(x+3)}{(x+4)(x-2)} \times \frac{(x+4)}{(x+4)} \times \frac{(x-2)}{(x-2)} = \frac{5x(x+3)(x+4)(x-2)}{(x+4)^2(x-2)^2}

However, we can simplify this expression further by canceling out the $(x+4)$ and $(x-2)$ terms in the numerator and denominator:

\frac{5x(x+3)(x+4)(x-2)}{(x+4)^2(x-2)^2} = \frac{5x(x+3)}{(x+4)}

Conclusion

In this article, we have simplified a complex algebraic expression involving fractions and variables. We have broken down the expression into manageable steps, factorized the numerators and denominators, canceled out common factors, and multiplied the numerators and denominators. The final simplified expression is $\frac{5x(x+3)}{(x+4)}$ .

Answer

Introduction

In our previous article, we explored how to simplify a complex algebraic expression involving fractions and variables. We broke down the expression into manageable steps, factorized the numerators and denominators, canceled out common factors, and multiplied the numerators and denominators. In this article, we will answer some common questions related to simplifying complex algebraic expressions.

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

A: The first step in simplifying a complex algebraic expression is to factorize the numerators and denominators. This involves breaking down the expression into simpler components, such as factoring out common terms or using the difference of squares formula.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you can use the following steps:

Look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term.
Write the middle term as the sum of two terms, each of which is a multiple of one of the numbers found in step 1.
Factor out the greatest common factor (GCF) of the two terms.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

a^2 - b^2 = (a - b)(a + b)

This formula can be used to factorize expressions of the form $a^2 - b^2$ .

Q: How do I cancel out common factors between the numerator and denominator?

A: To cancel out common factors between the numerator and denominator, you need to identify the common factors and divide both the numerator and denominator by the greatest common factor (GCF).

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides both the numerator and denominator of an expression.

Q: How do I multiply the numerators and denominators?

A: To multiply the numerators and denominators, you need to multiply each term in the numerator by each term in the denominator.

Q: What is the final step in simplifying a complex algebraic expression?

A: The final step in simplifying a complex algebraic expression is to simplify the expression further by canceling out any remaining common factors between the numerator and denominator.

Q: What are some common mistakes to avoid when simplifying complex algebraic expressions?

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

Not factoring out common terms
Not using the difference of squares formula
Not canceling out common factors between the numerator and denominator
Not multiplying the numerators and denominators correctly

Conclusion

In this article, we have answered some common questions related to simplifying complex algebraic expressions. We have covered topics such as factorizing quadratic expressions, using the difference of squares formula, canceling out common factors, multiplying the numerators and denominators, and avoiding common mistakes. By following these steps and avoiding common mistakes, you can simplify complex algebraic expressions with confidence.

Additional Resources

For more information on simplifying complex algebraic expressions, you can refer to the following resources:

Khan Academy: Algebra
Mathway: Algebra
Wolfram Alpha: Algebra

Practice Problems

To practice simplifying complex algebraic expressions, try the following problems:

Simplify the expression $\frac{x^2+5x+6}{x^2+3x+2}$ .
Simplify the expression $\frac{x^2-4x+4}{x^2-2x-3}$ .
Simplify the expression $\frac{x^2+2x+1}{x^2+5x+6}$ .

Answer Key

$\frac{x+3}{x+2}$
$\frac{x-2}{x-3}$
$\frac{x+1}{x+6}$