Simplify These Rational Expressions:a) $\frac{3x^2 - 21x + 36}{2x^2 - 32} \times \frac{2x + 10}{6x - 18}$

Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will focus on simplifying a given rational expression, which involves multiplying two fractions. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors between the numerator and denominator. In this case, we are given a rational expression that involves multiplying two fractions.

The Rational Expression

The given rational expression is:

$\frac{3x^2 - 21x + 36}{2x^2 - 32} \times \frac{2x + 10}{6x - 18}$

Step 1: Factor the Numerator and Denominator

To simplify the rational expression, we need to factor the numerator and denominator of each fraction.

Factoring the Numerator and Denominator

The numerator of the first fraction can be factored as:

$3x^2 - 21x + 36 = (3x - 12)(x - 3)$

The denominator of the first fraction can be factored as:

$2x^2 - 32 = 2(x^2 - 16) = 2(x - 4)(x + 4)$

The numerator of the second fraction can be factored as:

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

The denominator of the second fraction can be factored as:

$6x - 18 = 6(x - 3)$

Factored Form of the Rational Expression

Now that we have factored the numerator and denominator of each fraction, we can rewrite the rational expression in its factored form:

$\frac{(3x - 12)(x - 3)}{2(x - 4)(x + 4)} \times \frac{2(x + 5)}{6(x - 3)}$

Step 2: Cancel Out Common Factors

Now that we have the rational expression in its factored form, we can cancel out common factors between the numerator and denominator.

Canceling Out Common Factors

We can cancel out the common factor $(x - 3)$ between the numerator and denominator:

$\frac{(3x - 12)(x - 3)}{2(x - 4)(x + 4)} \times \frac{2(x + 5)}{6(x - 3)} = \frac{(3x - 12)}{2(x - 4)(x + 4)} \times \frac{2(x + 5)}{6}$

We can also cancel out the common factor $2$ between the numerator and denominator:

$\frac{(3x - 12)}{2(x - 4)(x + 4)} \times \frac{2(x + 5)}{6} = \frac{(3x - 12)(x + 5)}{6(x - 4)(x + 4)}$

Step 3: Simplify the Rational Expression

Now that we have canceled out common factors, we can simplify the rational expression by multiplying the remaining factors.

Simplifying the Rational Expression

We can simplify the rational expression by multiplying the remaining factors:

$\frac{(3x - 12)(x + 5)}{6(x - 4)(x + 4)} = \frac{(3x - 12)(x + 5)}{6(x - 4)(x + 4)}$

Conclusion

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, we can simplify a given rational expression by canceling out common factors and multiplying the remaining factors. Remember to always factor the numerator and denominator of each fraction and cancel out common factors before simplifying the rational expression.

Final Answer

The simplified rational expression is:

$\frac{(3x - 12)(x + 5)}{6(x - 4)(x + 4)}$

Example Use Case

Suppose we want to find the value of the rational expression when $x = 5$. We can plug in $x = 5$ into the simplified rational expression:

$\frac{(3(5) - 12)(5 + 5)}{6(5 - 4)(5 + 4)} = \frac{(15 - 12)(10)}{6(1)(9)} = \frac{3(10)}{54} = \frac{30}{54} = \frac{5}{9}$

Therefore, the value of the rational expression when $x = 5$ is $\frac{5}{9}$.

Tips and Tricks

• Always factor the numerator and denominator of each fraction before simplifying the rational expression.
• Cancel out common factors between the numerator and denominator before simplifying the rational expression.
• Multiply the remaining factors to simplify the rational expression.
• Use the example use case to practice simplifying rational expressions.

Common Mistakes

• Failing to factor the numerator and denominator of each fraction.
• Failing to cancel out common factors between the numerator and denominator.
• Failing to multiply the remaining factors to simplify the rational expression.

Conclusion

Introduction

In our previous article, we discussed how to simplify rational expressions by canceling out common factors and multiplying the remaining factors. However, we know that practice makes perfect, and the best way to learn is by doing. In this article, we will provide a Q&A guide to help you practice simplifying rational expressions.

Q1: What is the first step in simplifying a rational expression?

A1: The first step in simplifying a rational expression is to factor the numerator and denominator of each fraction.

Q2: How do I factor the numerator and denominator of each fraction?

A2: To factor the numerator and denominator of each fraction, you need to find the greatest common factor (GCF) of the terms and then factor out the GCF.

Q3: What is the greatest common factor (GCF)?

A3: The greatest common factor (GCF) is the largest factor that divides each term in a set of terms without leaving a remainder.

Q4: How do I cancel out common factors between the numerator and denominator?

A4: To cancel out common factors between the numerator and denominator, you need to identify the common factors and then cancel them out.

Q5: What is the final step in simplifying a rational expression?

A5: The final step in simplifying a rational expression is to multiply the remaining factors.

Q6: How do I multiply the remaining factors?

A6: To multiply the remaining factors, you need to multiply the numerators and denominators separately.

Q7: What are some common mistakes to avoid when simplifying rational expressions?

A7: Some common mistakes to avoid when simplifying rational expressions include failing to factor the numerator and denominator of each fraction, failing to cancel out common factors between the numerator and denominator, and failing to multiply the remaining factors.

Q8: How can I practice simplifying rational expressions?

A8: You can practice simplifying rational expressions by working through example problems and using online resources such as Khan Academy and Mathway.

Q9: What are some real-world applications of simplifying rational expressions?

A9: Simplifying rational expressions has many real-world applications, including solving equations, graphing functions, and modeling real-world phenomena.

Q10: How can I use technology to simplify rational expressions?

A10: You can use technology such as calculators and computer algebra systems to simplify rational expressions.

Conclusion

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article and practicing with example problems, you can become proficient in simplifying rational expressions. Remember to always factor the numerator and denominator of each fraction, cancel out common factors between the numerator and denominator, and multiply the remaining factors.

Example Problems

1. Simplify the rational expression: $\frac{x^2 + 5x + 6}{x^2 + 3x + 2}$
2. Simplify the rational expression: $\frac{2x^2 + 7x + 3}{x^2 + 2x + 1}$
3. Simplify the rational expression: $\frac{x^2 - 4x + 4}{x^2 - 2x + 1}$

Answer Key

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

Tips and Tricks

• Always factor the numerator and denominator of each fraction before simplifying the rational expression.
• Cancel out common factors between the numerator and denominator before simplifying the rational expression.
• Multiply the remaining factors to simplify the rational expression.
• Use the example use case to practice simplifying rational expressions.

Common Mistakes

• Failing to factor the numerator and denominator of each fraction.
• Failing to cancel out common factors between the numerator and denominator.
• Failing to multiply the remaining factors to simplify the rational expression.

Conclusion

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article and practicing with example problems, you can become proficient in simplifying rational expressions. Remember to always factor the numerator and denominator of each fraction, cancel out common factors between the numerator and denominator, and multiply the remaining factors.