Simplify $\frac{9x^2 + 15x + 4}{9x^2 - 1}$.

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved in simplifying rational expressions. In this article, we will focus on simplifying the given rational expression $\frac{9x^2 + 15x + 4}{9x^2 - 1}$.

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 factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression.

Factoring the Numerator and Denominator

To simplify the given rational expression, we need to factor the numerator and denominator.

Factoring the Numerator

The numerator of the given rational expression is $9x^2 + 15x + 4$. We can factor this quadratic expression by finding two numbers whose product is 36 (the product of the coefficient of the quadratic term and the constant term) and whose sum is 15 (the coefficient of the linear term).

import sympy as sp

x = sp.symbols('x')



numerator = 9x**2 + 15x + 4



factored_numerator = sp.factor(numerator)


print(factored_numerator)

The factored form of the numerator is $(3x + 4)(3x + 1)$.

Factoring the Denominator

The denominator of the given rational expression is $9x^2 - 1$. We can factor this quadratic expression by finding two numbers whose product is -9 (the product of the coefficient of the quadratic term and the constant term) and whose sum is 0 (the coefficient of the linear term).

# Define the denominator
denominator = 9*x**2 - 1


factored_denominator = sp.factor(denominator)


print(factored_denominator)

The factored form of the denominator is $(3x - 1)(3x + 1)$.

Canceling Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out common factors.

# Define the numerator and denominator
numerator = (3*x + 4)*(3*x + 1)
denominator = (3*x - 1)*(3*x + 1)


simplified_expression = sp.cancel(numerator/denominator)


print(simplified_expression)

The simplified form of the rational expression is $\frac{3x + 4}{3x - 1}$.

Conclusion

In this article, we simplified the given rational expression $\frac{9x^2 + 15x + 4}{9x^2 - 1}$ by factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression. The simplified form of the rational expression is $\frac{3x + 4}{3x - 1}$.

Final Answer

The final answer is $\boxed{\frac{3x + 4}{3x - 1}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

  1. Factor the numerator: $(3x + 4)(3x + 1)$
  2. Factor the denominator: $(3x - 1)(3x + 1)$
  3. Cancel out common factors: $\frac{(3x + 4)(3x + 1)}{(3x - 1)(3x + 1)}$
  4. Simplify the resulting expression: $\frac{3x + 4}{3x - 1}$

Frequently Asked Questions

Q: What is the simplified form of the rational expression $\frac{9x^2 + 15x + 4}{9x^2 - 1}$?

A: The simplified form of the rational expression is $\frac{3x + 4}{3x - 1}$.

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 then simplify the resulting expression.

Q: What is the final answer to the problem?

A: The final answer is $\boxed{\frac{3x + 4}{3x - 1}}$.

References

Related Topics

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Introduction

In our previous article, we simplified the given rational expression $\frac{9x^2 + 15x + 4}{9x^2 - 1}$ by factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression. In this article, we will provide a Q&A section to help you better understand the concept of simplifying rational expressions.

Q&A

Q: What is a rational expression?

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

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 then simplify the resulting expression.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves breaking down an expression into its prime factors, while canceling out common factors involves removing any common factors between the numerator and denominator.

Q: Can I simplify a rational expression if it has no common factors?

A: Yes, you can still simplify a rational expression even if it has no common factors. You can simplify it by combining like terms in the numerator and denominator.

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

A: You can check if a rational expression can be simplified by factoring the numerator and denominator and canceling out any common factors.

Q: What is the final answer to the problem?

A: The final answer is $\boxed{\frac{3x + 4}{3x - 1}}$.

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

A: Yes, you can use a calculator to simplify a rational expression. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: How do I check my work when simplifying a rational expression?

A: You can check your work by plugging in a value for the variable and making sure the expression simplifies to the correct value.

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

A: Some common mistakes to avoid when simplifying rational expressions include:

  • Not factoring the numerator and denominator
  • Not canceling out common factors
  • Not combining like terms in the numerator and denominator
  • Not checking your work by hand

Tips and Tricks

  • Always factor the numerator and denominator before simplifying a rational expression.
  • Make sure to cancel out any common factors between the numerator and denominator.
  • Combine like terms in the numerator and denominator to simplify the expression.
  • Check your work by hand to make sure you get the correct answer.

Common Misconceptions

  • Many students believe that simplifying rational expressions is only necessary when the numerator and denominator have common factors. However, simplifying rational expressions can also involve combining like terms in the numerator and denominator.
  • Some students believe that simplifying rational expressions is only necessary when the expression is in the form of a fraction. However, simplifying rational expressions can also involve simplifying expressions that are not in the form of a fraction.

Conclusion

In this article, we provided a Q&A section to help you better understand the concept of simplifying rational expressions. We also provided some tips and tricks to help you simplify rational expressions, as well as some common misconceptions to avoid. By following these tips and avoiding these misconceptions, you can become more confident in your ability to simplify rational expressions.

Final Answer

The final answer is $\boxed{\frac{3x + 4}{3x - 1}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

  1. Factor the numerator: $(3x + 4)(3x + 1)$
  2. Factor the denominator: $(3x - 1)(3x + 1)$
  3. Cancel out common factors: $\frac{(3x + 4)(3x + 1)}{(3x - 1)(3x + 1)}$
  4. Simplify the resulting expression: $\frac{3x + 4}{3x - 1}$

Frequently Asked Questions

Q: What is the simplified form of the rational expression $\frac{9x^2 + 15x + 4}{9x^2 - 1}$?

A: The simplified form of the rational expression is $\frac{3x + 4}{3x - 1}$.

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 then simplify the resulting expression.

Q: What is the final answer to the problem?

A: The final answer is $\boxed{\frac{3x + 4}{3x - 1}}$.

References

Related Topics