Simplify The Expression:${ \frac{(8v^3 + 74v^2 - 116v + 37)}{(8v - 6)} }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems more efficiently and accurately. One of the most common techniques used to simplify expressions is factoring, which involves breaking down an expression into simpler components. In this article, we will focus on simplifying the given expression using factoring and canceling techniques.

The Given Expression

The given expression is:

(8v3+74v2−116v+37)(8v−6)\frac{(8v^3 + 74v^2 - 116v + 37)}{(8v - 6)}

Our goal is to simplify this expression by factoring the numerator and canceling out any common factors between the numerator and denominator.

Factoring the Numerator

To factor the numerator, we need to find the greatest common factor (GCF) of the four terms. The GCF of 8, 74, -116, and 37 is 1, which means that there is no common factor among the four terms. However, we can try to factor the numerator by grouping the terms.

Let's group the first two terms and the last two terms:

(8v3+74v2)+(−116v+37)(8v−6)\frac{(8v^3 + 74v^2) + (-116v + 37)}{(8v - 6)}

Now, let's factor out the common factor from each group:

2v2(4v+37)−29(4v−1)(8v−6)\frac{2v^2(4v + 37) - 29(4v - 1)}{(8v - 6)}

Factoring the Denominator

The denominator is already factored as:

8v−68v - 6

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we can factor out a 2 from the numerator and a 2 from the denominator:

2v2(4v+37)−29(4v−1)2(4v−3)\frac{2v^2(4v + 37) - 29(4v - 1)}{2(4v - 3)}

Canceling the Common Factor

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

v2(4v+37)−29(4v−1)4v−3\frac{v^2(4v + 37) - 29(4v - 1)}{4v - 3}

Simplifying the Expression

Now that we have canceled out the common factor, we can simplify the expression further by combining like terms:

4v3+37v2−116v+294v−3\frac{4v^3 + 37v^2 - 116v + 29}{4v - 3}

Final Simplification

We can simplify the expression further by factoring the numerator:

(4v−3)(v2+4v+10)4v−3\frac{(4v - 3)(v^2 + 4v + 10)}{4v - 3}

Canceling the Common Factor

We can cancel out the common factor of (4v - 3) between the numerator and denominator:

v2+4v+10v^2 + 4v + 10

Conclusion

In this article, we simplified the given expression using factoring and canceling techniques. We factored the numerator and denominator, canceled out common factors, and simplified the expression further by combining like terms. The final simplified expression is:

v2+4v+10v^2 + 4v + 10

Tips and Tricks

  • When simplifying expressions, always look for common factors between the numerator and denominator.
  • Use factoring techniques to break down complex expressions into simpler components.
  • Cancel out common factors to simplify the expression further.
  • Combine like terms to simplify the expression even further.

Common Mistakes to Avoid

  • Failing to factor the numerator and denominator.
  • Not canceling out common factors.
  • Not combining like terms.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. For example, in physics, we use simplifying expressions to solve problems involving motion and energy. In engineering, we use simplifying expressions to design and optimize systems. In finance, we use simplifying expressions to calculate interest rates and investment returns.

Final Thoughts

Simplifying expressions is a powerful technique that can help us solve problems more efficiently and accurately. By factoring the numerator and denominator, canceling out common factors, and combining like terms, we can simplify complex expressions and arrive at a final solution. Remember to always look for common factors, use factoring techniques, cancel out common factors, and combine like terms to simplify expressions.

Introduction

In our previous article, we simplified the given expression using factoring and canceling techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to look for common factors between the numerator and denominator. If there are no common factors, we can try to factor the numerator and denominator.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, we need to find the greatest common factor (GCF) of the terms. If there is no common factor, we can try to factor the terms by grouping them.

Q: What is the difference between factoring and canceling?

A: Factoring involves breaking down an expression into simpler components, while canceling involves eliminating common factors between the numerator and denominator.

Q: Can I cancel out a common factor if it is not a factor of both the numerator and denominator?

A: No, you cannot cancel out a common factor if it is not a factor of both the numerator and denominator. Canceling is only possible if the common factor is a factor of both the numerator and denominator.

Q: How do I know if I have canceled out all the common factors?

A: To check if you have canceled out all the common factors, you can try to simplify the expression further by combining like terms.

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to check if the expression can be simplified further by combining like terms.

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

A: Yes, you can use a calculator to simplify an expression. However, it is always a good idea to check the answer by hand to ensure that it is correct.

Q: How do I know if an expression is already simplified?

A: An expression is already simplified if there are no common factors between the numerator and denominator, and the terms are combined in the simplest possible way.

Q: Can I simplify an expression that has a variable in the denominator?

A: Yes, you can simplify an expression that has a variable in the denominator. However, you need to be careful when canceling out common factors to avoid dividing by zero.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to factor the numerator and denominator separately and then cancel out common factors.

Q: Can I use a graphing calculator to simplify an expression?

A: Yes, you can use a graphing calculator to simplify an expression. However, it is always a good idea to check the answer by hand to ensure that it is correct.

Tips and Tricks

  • Always look for common factors between the numerator and denominator.
  • Use factoring techniques to break down complex expressions into simpler components.
  • Cancel out common factors to simplify the expression further.
  • Combine like terms to simplify the expression even further.
  • Check the answer by hand to ensure that it is correct.

Common Mistakes to Avoid

  • Failing to factor the numerator and denominator.
  • Not canceling out common factors.
  • Not combining like terms.
  • Dividing by zero.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. For example, in physics, we use simplifying expressions to solve problems involving motion and energy. In engineering, we use simplifying expressions to design and optimize systems. In finance, we use simplifying expressions to calculate interest rates and investment returns.

Final Thoughts

Simplifying expressions is a powerful technique that can help us solve problems more efficiently and accurately. By factoring the numerator and denominator, canceling out common factors, and combining like terms, we can simplify complex expressions and arrive at a final solution. Remember to always look for common factors, use factoring techniques, cancel out common factors, and combine like terms to simplify expressions.