Select The Correct Answer.Which Expression Is Equivalent To $\frac{10y^7 - 8y^6 + 3y 4}{y 3}$? Assume That The Denominator Does Not Equal Zero.A. $y 2(10y 3 - 8y^2 + 3)$ B. $ 10 Y 3 − 8 Y 2 + 3 Y \frac{10y^3 - 8y^2 + 3}{y} Y 10 Y 3 − 8 Y 2 + 3 ​ [/tex] C.

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore how to simplify a given algebraic expression, specifically the expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$. We will break down the expression into smaller parts, apply the rules of exponents, and finally arrive at the simplified form.

Understanding the Expression

The given expression is $\frac{10y^7 - 8y^6 + 3y4}{y3}$. This expression consists of a numerator and a denominator. The numerator is a polynomial expression with three terms, while the denominator is a single term, y3y^3. Our goal is to simplify this expression by canceling out any common factors between the numerator and the denominator.

Step 1: Factor Out the Greatest Common Factor (GCF)

The first step in simplifying the expression is to factor out the greatest common factor (GCF) from the numerator. The GCF of the three terms in the numerator is y4y^4. We can factor out y4y^4 from each term in the numerator:

10y7−8y6+3y4y3=y4(10y3−8y2+3)y3\frac{10y^7 - 8y^6 + 3y^4}{y^3} = \frac{y^4(10y^3 - 8y^2 + 3)}{y^3}

Step 2: Cancel Out Common Factors

Now that we have factored out the GCF, we can cancel out any common factors between the numerator and the denominator. In this case, we can cancel out one factor of y3y^3 from the numerator and the denominator:

y4(10y3−8y2+3)y3=y4(10y3−8y2+3)\frac{y^4(10y^3 - 8y^2 + 3)}{y^3} = y^4(10y^3 - 8y^2 + 3)

However, we are not done yet. We can simplify the expression further by canceling out one factor of y4y^4 from the numerator and the denominator:

y4(10y3−8y2+3)=y2(10y3−8y2+3)y^4(10y^3 - 8y^2 + 3) = y^2(10y^3 - 8y^2 + 3)

Conclusion

In conclusion, the simplified form of the expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$ is y2(10y3−8y2+3)y^2(10y^3 - 8y^2 + 3). This expression is equivalent to the original expression, and it is the simplest form of the expression.

Answer

The correct answer is:

  • A. y2(10y3−8y2+3)y^2(10y^3 - 8y^2 + 3)

Discussion

This problem requires the application of the rules of exponents and the concept of factoring out the greatest common factor (GCF). The expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$ can be simplified by factoring out the GCF and canceling out common factors between the numerator and the denominator. The simplified form of the expression is y2(10y3−8y2+3)y^2(10y^3 - 8y^2 + 3).

Example Use Cases

This problem can be used as an example in a math class to demonstrate the concept of simplifying algebraic expressions. The problem can be used to illustrate the importance of factoring out the GCF and canceling out common factors between the numerator and the denominator.

Tips and Variations

  • To make the problem more challenging, you can add more terms to the numerator or denominator.
  • To make the problem easier, you can simplify the expression by canceling out more common factors between the numerator and the denominator.
  • You can also use this problem as a starting point to explore other concepts in algebra, such as polynomial division or rational expressions.
    Simplifying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored how to simplify a given algebraic expression, specifically the expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$. We broke down the expression into smaller parts, applied the rules of exponents, and finally arrived at the simplified form. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

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

A: The greatest common factor (GCF) is the largest factor that divides each term in a polynomial expression. In the expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$, the GCF is y4y^4.

Q: How do I factor out the GCF from a polynomial expression?

A: To factor out the GCF from a polynomial expression, you need to identify the GCF and then multiply each term in the expression by the GCF. For example, in the expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$, we can factor out y4y^4 from each term:

10y7−8y6+3y4y3=y4(10y3−8y2+3)y3\frac{10y^7 - 8y^6 + 3y^4}{y^3} = \frac{y^4(10y^3 - 8y^2 + 3)}{y^3}

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

A: Factoring out the GCF involves multiplying each term in the expression by the GCF, while canceling out common factors involves dividing the numerator and denominator by a common factor. In the expression $\frac{10y^7 - 8y^6 + 3y4}{y3}$, we can factor out y4y^4 from each term and then cancel out one factor of y3y^3 from the numerator and the denominator:

y4(10y3−8y2+3)y3=y4(10y3−8y2+3)\frac{y^4(10y^3 - 8y^2 + 3)}{y^3} = y^4(10y^3 - 8y^2 + 3)

Q: How do I know when to factor out the GCF and when to cancel out common factors?

A: You should factor out the GCF when the expression has multiple terms and you want to simplify it by identifying the largest factor that divides each term. You should cancel out common factors when the expression has a common factor between the numerator and the denominator.

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

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

  • Not factoring out the GCF correctly
  • Canceling out common factors incorrectly
  • Not simplifying the expression fully
  • Not checking for any remaining common factors between the numerator and the denominator

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By understanding the concept of factoring out the greatest common factor (GCF) and canceling out common factors, you can simplify complex expressions and arrive at the simplest form. Remember to identify the GCF, factor it out, and then cancel out common factors to simplify the expression.

Tips and Variations

  • To make the problem more challenging, you can add more terms to the numerator or denominator.
  • To make the problem easier, you can simplify the expression by canceling out more common factors between the numerator and the denominator.
  • You can also use this problem as a starting point to explore other concepts in algebra, such as polynomial division or rational expressions.

Example Use Cases

This problem can be used as an example in a math class to demonstrate the concept of simplifying algebraic expressions. The problem can be used to illustrate the importance of factoring out the GCF and canceling out common factors between the numerator and the denominator.

Practice Problems

Here are some practice problems to help you reinforce your understanding of simplifying algebraic expressions:

  1. Simplify the expression $\frac{12x^5 - 9x^4 + 6x2}{x2}$.
  2. Simplify the expression $\frac{15y^8 - 12y^7 + 9y5}{y3}$.
  3. Simplify the expression $\frac{20z^9 - 15z^8 + 10z6}{z4}$.

Answer Key

  1. 12x3−9x2+612x^3 - 9x^2 + 6

  2. 15y5−12y4+9y215y^5 - 12y^4 + 9y^2

  3. 20z5−15z4+10z220z^5 - 15z^4 + 10z^2