What Is The Factored Form Of The Polynomial $27x^2y - 43xy^2$?A. $xy(27x - 43y)$ B. \$3x^2y(9 - 14y)$[/tex\] C. $3xy(9x - 17y)$ D. $x^2y^2(27 - 43)$

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Introduction

Factoring polynomials is an essential skill in algebra, and it's used to simplify complex expressions and solve equations. In this article, we'll explore the factored form of the polynomial $27x^2y - 43xy^2$ and examine the different options provided.

Understanding the Polynomial

The given polynomial is $27x^2y - 43xy^2$. To factor this expression, we need to identify the greatest common factor (GCF) of the two terms. The GCF is the largest expression that divides both terms without leaving a remainder.

Factoring the Polynomial

To factor the polynomial, we can start by identifying the GCF of the two terms. In this case, the GCF is $xy$. We can factor out $xy$ from both terms:

27x2y−43xy2=xy(27x−43y)27x^2y - 43xy^2 = xy(27x - 43y)

Examining the Options

Now that we have factored the polynomial, let's examine the options provided:

  • A. $xy(27x - 43y)$
  • B. $3x^2y(9 - 14y)$
  • C. $3xy(9x - 17y)$
  • D. $x2y2(27 - 43)$

Analyzing Option A

Option A is $xy(27x - 43y)$. This option matches the factored form we obtained earlier. The GCF of the two terms is indeed $xy$, and the remaining expression is $27x - 43y$.

Analyzing Option B

Option B is $3x^2y(9 - 14y)$. This option does not match the factored form we obtained earlier. The GCF of the two terms is not $3x^2y$, and the remaining expression is not $9 - 14y$.

Analyzing Option C

Option C is $3xy(9x - 17y)$. This option does not match the factored form we obtained earlier. The GCF of the two terms is not $3xy$, and the remaining expression is not $9x - 17y$.

Analyzing Option D

Option D is $x2y2(27 - 43)$. This option does not match the factored form we obtained earlier. The GCF of the two terms is not $x2y2$, and the remaining expression is not $27 - 43$.

Conclusion

In conclusion, the factored form of the polynomial $27x^2y - 43xy^2$ is $xy(27x - 43y)$. This matches option A, which is the correct answer.

Final Answer

The final answer is A. $xy(27x - 43y)$.

Frequently Asked Questions

  • Q: What is the greatest common factor (GCF) of the two terms in the polynomial $27x^2y - 43xy^2$? A: The GCF is $xy$.
  • Q: How do you factor the polynomial $27x^2y - 43xy^2$? A: To factor the polynomial, you can identify the GCF of the two terms and factor it out.
  • Q: What is the factored form of the polynomial $27x^2y - 43xy^2$? A: The factored form is $xy(27x - 43y)$.

Additional Resources

  • For more information on factoring polynomials, visit the Khan Academy website.
  • For more practice problems on factoring polynomials, visit the Mathway website.

Related Topics

  • Factoring quadratic expressions
  • Factoring polynomial expressions
  • Greatest common factor (GCF)

References

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak
  • "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Introduction

Factoring polynomials is an essential skill in algebra, and it's used to simplify complex expressions and solve equations. In this article, we'll provide a Q&A guide to help you understand the concept of factoring polynomials and how to apply it to different types of expressions.

Q: What is factoring a polynomial?

A: Factoring a polynomial is the process of expressing it as a product of simpler expressions, called factors. This is done by identifying the greatest common factor (GCF) of the terms and factoring it out.

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

A: The greatest common factor (GCF) is the largest expression that divides both terms without leaving a remainder. It's the product of the common factors of the terms.

Q: How do you factor a polynomial?

A: To factor a polynomial, you need to identify the GCF of the terms and factor it out. You can do this by:

  • Identifying the common factors of the terms
  • Factoring out the GCF
  • Writing the remaining expression as a product of simpler expressions

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Factoring out the GCF
  • Factoring by grouping
  • Factoring quadratic expressions
  • Factoring polynomial expressions

Q: How do you factor out the GCF?

A: To factor out the GCF, you need to identify the GCF of the terms and factor it out. You can do this by:

  • Identifying the common factors of the terms
  • Factoring out the GCF
  • Writing the remaining expression as a product of simpler expressions

Q: How do you factor by grouping?

A: To factor by grouping, you need to group the terms into pairs and factor out the GCF of each pair. You can do this by:

  • Grouping the terms into pairs
  • Factoring out the GCF of each pair
  • Writing the remaining expression as a product of simpler expressions

Q: How do you factor quadratic expressions?

A: To factor quadratic expressions, you need to identify the two binomials that multiply to give the quadratic expression. You can do this by:

  • Identifying the two binomials
  • Factoring the quadratic expression as a product of the two binomials

Q: How do you factor polynomial expressions?

A: To factor polynomial expressions, you need to identify the GCF of the terms and factor it out. You can do this by:

  • Identifying the common factors of the terms
  • Factoring out the GCF
  • Writing the remaining expression as a product of simpler expressions

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

  • Not identifying the GCF correctly
  • Factoring out the wrong expression
  • Not writing the remaining expression as a product of simpler expressions

Q: How do you check if a factored form is correct?

A: To check if a factored form is correct, you need to multiply the factors together and see if you get the original expression. You can do this by:

  • Multiplying the factors together
  • Checking if the result is equal to the original expression

Q: What are some real-world applications of factoring polynomials?

A: Factoring polynomials has many real-world applications, including:

  • Simplifying complex expressions in physics and engineering
  • Solving equations in computer science and cryptography
  • Modeling population growth and decay in biology and economics

Conclusion

In conclusion, factoring polynomials is an essential skill in algebra, and it's used to simplify complex expressions and solve equations. By understanding the concept of factoring polynomials and how to apply it to different types of expressions, you can become proficient in this skill and apply it to real-world problems.

Final Answer

The final answer is that factoring polynomials is a powerful tool for simplifying complex expressions and solving equations.

Frequently Asked Questions

  • Q: What is factoring a polynomial? A: Factoring a polynomial is the process of expressing it as a product of simpler expressions, called factors.
  • Q: What is the greatest common factor (GCF)? A: The greatest common factor (GCF) is the largest expression that divides both terms without leaving a remainder.
  • Q: How do you factor a polynomial? A: To factor a polynomial, you need to identify the GCF of the terms and factor it out.

Additional Resources

  • For more information on factoring polynomials, visit the Khan Academy website.
  • For more practice problems on factoring polynomials, visit the Mathway website.

Related Topics

  • Factoring quadratic expressions
  • Factoring polynomial expressions
  • Greatest common factor (GCF)

References

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak
  • "Mathematics for Computer Science" by Eric Lehman and Tom Leighton