Simplify The Expression: 7 X − 1 6 − 5 X + X 2 \frac{7x-1}{6-5x+x^2} 6 − 5 X + X 2 7 X − 1 ​

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One common technique used to simplify expressions is rationalizing the denominator. In this article, we will focus on simplifying the expression 7x−16−5x+x2\frac{7x-1}{6-5x+x^2} by rationalizing the denominator.

Understanding the Expression

The given expression is 7x−16−5x+x2\frac{7x-1}{6-5x+x^2}. To simplify this expression, we need to understand the concept of rationalizing the denominator. Rationalizing the denominator involves multiplying both the numerator and the denominator by a cleverly chosen expression to eliminate any radicals or complex numbers in the denominator.

Step 1: Factor the Denominator

The first step in simplifying the expression is to factor the denominator. The denominator is 6−5x+x26-5x+x^2, which can be factored as (3−x)(2+x)(3-x)(2+x). This is a difference of squares, where 6−5x+x2=(3−x)2+(2+x)26-5x+x^2 = (3-x)^2 + (2+x)^2.

Step 2: Multiply by the Conjugate

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (3−x)(2+x)(3-x)(2+x) is (3−x)(2+x)(3-x)(2+x). By multiplying both the numerator and the denominator by this expression, we can eliminate the radicals in the denominator.

Step 3: Simplify the Expression

Now that we have multiplied both the numerator and the denominator by the conjugate, we can simplify the expression. The expression becomes:

(7x−1)(3−x)(2+x)(3−x)2(2+x)2\frac{(7x-1)(3-x)(2+x)}{(3-x)^2(2+x)^2}

Step 4: Cancel Common Factors

The next step is to cancel any common factors between the numerator and the denominator. In this case, we can cancel the (3−x)(3-x) and (2+x)(2+x) factors.

Step 5: Simplify the Expression Further

After canceling the common factors, the expression becomes:

7x−1(3−x)(2+x)\frac{7x-1}{(3-x)(2+x)}

Step 6: Final Simplification

The final step is to simplify the expression further by factoring the denominator. The denominator (3−x)(2+x)(3-x)(2+x) can be factored as (3−x)(2+x)=6−5x+x2(3-x)(2+x) = 6-5x+x^2. This is the same expression we had in the original problem.

Conclusion

In conclusion, we have successfully simplified the expression 7x−16−5x+x2\frac{7x-1}{6-5x+x^2} by rationalizing the denominator. The final simplified expression is 7x−16−5x+x2\frac{7x-1}{6-5x+x^2}. This expression cannot be simplified further.

Tips and Tricks

  • When rationalizing the denominator, always multiply both the numerator and the denominator by the conjugate of the denominator.
  • Make sure to cancel any common factors between the numerator and the denominator.
  • Factor the denominator to simplify the expression further.

Real-World Applications

Rationalizing the denominator is a crucial technique used in many real-world applications, such as:

  • Engineering: Rationalizing the denominator is used to simplify complex expressions in engineering problems, such as electrical circuits and mechanical systems.
  • Physics: Rationalizing the denominator is used to simplify complex expressions in physics problems, such as wave equations and quantum mechanics.
  • Computer Science: Rationalizing the denominator is used to simplify complex expressions in computer science problems, such as algorithm design and data analysis.

Common Mistakes

  • Failing to multiply both the numerator and the denominator by the conjugate of the denominator.
  • Failing to cancel common factors between the numerator and the denominator.
  • Not factoring the denominator to simplify the expression further.

Conclusion

In conclusion, simplifying the expression 7x−16−5x+x2\frac{7x-1}{6-5x+x^2} by rationalizing the denominator is a crucial skill that helps us solve equations and inequalities. By following the steps outlined in this article, we can successfully simplify the expression and arrive at the final simplified expression. Remember to always multiply both the numerator and the denominator by the conjugate of the denominator, cancel common factors, and factor the denominator to simplify the expression further.

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Introduction

In our previous article, we discussed how to simplify the expression 7x−16−5x+x2\frac{7x-1}{6-5x+x^2} by rationalizing the denominator. In this article, we will provide a Q&A guide to help you understand the concept of rationalizing the denominator and how to apply it to simplify expressions.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a technique used to simplify expressions by eliminating any radicals or complex numbers in the denominator. It involves multiplying both the numerator and the denominator by a cleverly chosen expression to eliminate the radicals or complex numbers.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify expressions and make them easier to work with. Rationalizing the denominator helps us to eliminate any radicals or complex numbers in the denominator, which can make it easier to solve equations and inequalities.

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator is an expression that is identical to the denominator, but with the opposite sign.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is an expression that is identical to the denominator, but with the opposite sign. For example, if the denominator is a−ba-b, the conjugate is a+ba+b.

Q: How do we multiply the numerator and the denominator by the conjugate?

A: To multiply the numerator and the denominator by the conjugate, we need to multiply both expressions by the conjugate. This will eliminate any radicals or complex numbers in the denominator.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Failing to multiply both the numerator and the denominator by the conjugate of the denominator.
  • Failing to cancel common factors between the numerator and the denominator.
  • Not factoring the denominator to simplify the expression further.

Q: How do we simplify the expression after rationalizing the denominator?

A: After rationalizing the denominator, we need to simplify the expression by canceling any common factors between the numerator and the denominator. We also need to factor the denominator to simplify the expression further.

Q: What are some real-world applications of rationalizing the denominator?

A: Rationalizing the denominator has many real-world applications, including:

  • Engineering: Rationalizing the denominator is used to simplify complex expressions in engineering problems, such as electrical circuits and mechanical systems.
  • Physics: Rationalizing the denominator is used to simplify complex expressions in physics problems, such as wave equations and quantum mechanics.
  • Computer Science: Rationalizing the denominator is used to simplify complex expressions in computer science problems, such as algorithm design and data analysis.

Tips and Tricks

  • Always multiply both the numerator and the denominator by the conjugate of the denominator.
  • Make sure to cancel any common factors between the numerator and the denominator.
  • Factor the denominator to simplify the expression further.
  • Use the conjugate of the denominator to eliminate any radicals or complex numbers in the denominator.

Conclusion

In conclusion, rationalizing the denominator is a crucial technique used to simplify expressions and make them easier to work with. By following the steps outlined in this article, you can successfully simplify expressions and arrive at the final simplified expression. Remember to always multiply both the numerator and the denominator by the conjugate of the denominator, cancel common factors, and factor the denominator to simplify the expression further.

Frequently Asked Questions

Q: What is the difference between rationalizing the denominator and simplifying the expression?

A: Rationalizing the denominator is a technique used to eliminate any radicals or complex numbers in the denominator, while simplifying the expression involves canceling any common factors between the numerator and the denominator.

Q: Can I use rationalizing the denominator to simplify expressions with rational exponents?

A: Yes, you can use rationalizing the denominator to simplify expressions with rational exponents. However, you need to be careful when dealing with rational exponents, as they can be tricky to work with.

Q: Can I use rationalizing the denominator to simplify expressions with complex numbers?

A: Yes, you can use rationalizing the denominator to simplify expressions with complex numbers. However, you need to be careful when dealing with complex numbers, as they can be tricky to work with.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Failing to multiply both the numerator and the denominator by the conjugate of the denominator.
  • Failing to cancel common factors between the numerator and the denominator.
  • Not factoring the denominator to simplify the expression further.

Conclusion

In conclusion, rationalizing the denominator is a crucial technique used to simplify expressions and make them easier to work with. By following the steps outlined in this article, you can successfully simplify expressions and arrive at the final simplified expression. Remember to always multiply both the numerator and the denominator by the conjugate of the denominator, cancel common factors, and factor the denominator to simplify the expression further.