Simplify The Rational Expression And Find All Numbers That Must Be Excluded From The Domain Of The Simplified Rational Expression. 4 X + 2 12 X 2 + 26 X + 10 \frac{4x + 2}{12x^2 + 26x + 10} 12 X 2 + 26 X + 10 4 X + 2 ​

Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial step in solving equations and inequalities. In this article, we will simplify the rational expression $\frac{4x + 2}{12x^2 + 26x + 10}$ and find all numbers that must be excluded from the domain of the simplified rational expression.

What is a Rational Expression?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression.

Simplifying the Rational Expression

To simplify the rational expression $\frac{4x + 2}{12x^2 + 26x + 10}$, we need to factor the numerator and denominator.

Factoring the Numerator

The numerator $4x + 2$ can be factored as $2(2x + 1)$.

Factoring the Denominator

The denominator $12x^2 + 26x + 10$ can be factored as $(2x + 1)(6x + 5)$.

Simplifying the Rational Expression

Now that we have factored the numerator and denominator, we can simplify the rational expression by canceling out the common factor $(2x + 1)$.

$$\frac{4x + 2}{12x^2 + 26x + 10} = \frac{2(2x + 1)}{(2x + 1)(6x + 5)} = \frac{2}{6x + 5}$$

Excluding Numbers from the Domain

The domain of a rational expression is the set of all real numbers that can be plugged into the expression without resulting in an undefined value. In the case of the simplified rational expression $\frac{2}{6x + 5}$, the denominator $6x + 5$ cannot be equal to zero, as this would result in an undefined value.

Finding the Numbers to Exclude

To find the numbers that must be excluded from the domain, we need to solve the equation $6x + 5 = 0$.

$$6x + 5 = 0$$

Subtracting 5 from both sides gives:

$$6x = -5$$

Dividing both sides by 6 gives:

$$x = -\frac{5}{6}$$

Therefore, the number that must be excluded from the domain is $x = -\frac{5}{6}$.

Conclusion

In this article, we simplified the rational expression $\frac{4x + 2}{12x^2 + 26x + 10}$ and found the number that must be excluded from the domain of the simplified rational expression. The simplified rational expression is $\frac{2}{6x + 5}$, and the number that must be excluded from the domain is $x = -\frac{5}{6}$.

Final Answer

The final answer is $\boxed{-\frac{5}{6}}$.

Additional Tips and Tricks

• When simplifying rational expressions, always factor the numerator and denominator.
• Cancel out common factors between the numerator and denominator.
• Be careful when excluding numbers from the domain, as this can affect the validity of the expression.

Common Mistakes to Avoid

• Failing to factor the numerator and denominator.
• Not canceling out common factors between the numerator and denominator.
• Excluding numbers from the domain without checking for validity.

Real-World Applications

Rational expressions have numerous real-world applications, including:

• Physics and Engineering: Rational expressions are used to model real-world phenomena, such as motion and energy.
• Economics: Rational expressions are used to model economic systems and make predictions about future trends.
• Computer Science: Rational expressions are used in computer algorithms and programming languages.

Conclusion

Introduction

In our previous article, we simplified the rational expression $\frac{4x + 2}{12x^2 + 26x + 10}$ and found the number that must be excluded from the domain of the simplified rational expression. In this article, we will answer some frequently asked questions about simplifying rational expressions and excluding numbers from the domain.

Q&A

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

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. A fraction is a specific type of rational expression where the numerator and denominator are integers.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out common factors, and simplify the resulting expression.

Q: What is the domain of a rational expression?

A: The domain of a rational expression is the set of all real numbers that can be plugged into the expression without resulting in an undefined value.

Q: How do I find the numbers to exclude from the domain?

A: To find the numbers to exclude from the domain, you need to solve the equation that makes the denominator equal to zero.

Q: What is the final answer for the simplified rational expression $\frac{4x + 2}{12x^2 + 26x + 10}$?

A: The final answer is $\boxed{-\frac{5}{6}}$.

Q: Can I use a calculator to simplify rational expressions?

A: Yes, you can use a calculator to simplify rational expressions, but it's always a good idea to check your work by hand to ensure accuracy.

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

A: A rational expression is simplified when there are no common factors between the numerator and denominator.

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, but you need to be careful when excluding numbers from the domain.

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

A: A rational expression is a type of algebraic expression that contains variables and/or constants in the numerator and/or denominator.

Q: Can I use rational expressions to solve equations and inequalities?

A: Yes, you can use rational expressions to solve equations and inequalities, but you need to be careful when excluding numbers from the domain.

Conclusion

In conclusion, simplifying rational expressions and excluding numbers from the domain is a crucial step in solving equations and inequalities. By following the tips and tricks outlined in this article, you can simplify rational expressions and find the numbers that must be excluded from the domain. Remember to always factor the numerator and denominator, cancel out common factors, and be careful when excluding numbers from the domain.

Additional Tips and Tricks

• When simplifying rational expressions, always factor the numerator and denominator.
• Cancel out common factors between the numerator and denominator.
• Be careful when excluding numbers from the domain, as this can affect the validity of the expression.
• Use a calculator to check your work and ensure accuracy.
• Simplify rational expressions with variables in the denominator carefully.

Common Mistakes to Avoid

• Failing to factor the numerator and denominator.
• Not canceling out common factors between the numerator and denominator.
• Excluding numbers from the domain without checking for validity.
• Using a calculator without checking your work by hand.
• Simplifying rational expressions with variables in the denominator without careful consideration.

Real-World Applications

Rational expressions have numerous real-world applications, including:

• Physics and Engineering: Rational expressions are used to model real-world phenomena, such as motion and energy.
• Economics: Rational expressions are used to model economic systems and make predictions about future trends.
• Computer Science: Rational expressions are used in computer algorithms and programming languages.

Conclusion

In conclusion, simplifying rational expressions and excluding numbers from the domain is a crucial step in solving equations and inequalities. By following the tips and tricks outlined in this article, you can simplify rational expressions and find the numbers that must be excluded from the domain. Remember to always factor the numerator and denominator, cancel out common factors, and be careful when excluding numbers from the domain.