Factor The Expression Given Below. Write Each Factor As A Polynomial In Descending Order. Enter Exponents Using The Caret ( ^ ). For Example, You Would Enter $x^2$ As $x^{\wedge} 2$.$ 125 X 3 + 343 Y 3 125x^3 + 343y^3 125 X 3 + 343 Y 3 [/tex]

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Introduction

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the given expression 125x^3 + 343y^3. We will use the concept of the sum of cubes to factor this expression and express each factor as a polynomial in descending order.

Understanding the Sum of Cubes

The sum of cubes is a mathematical formula that states:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

This formula can be used to factor expressions of the form a^3 + b^3. In our given expression, 125x^3 + 343y^3, we can see that it can be written as (5x)^3 + (7y)^3, which is in the form of a^3 + b^3.

Factoring the Expression

Using the sum of cubes formula, we can factor the expression 125x^3 + 343y^3 as follows:

125x^3 + 343y^3 = (5x)^3 + (7y)^3 = (5x + 7y)((5x)^2 - (5x)(7y) + (7y)^2) = (5x + 7y)(25x^2 - 35xy + 49y^2)

Expressing Each Factor as a Polynomial in Descending Order

In the factored form of the expression, we have two factors: (5x + 7y) and (25x^2 - 35xy + 49y^2). We need to express each factor as a polynomial in descending order.

The first factor, (5x + 7y), is already in descending order.

The second factor, (25x^2 - 35xy + 49y^2), can be expressed in descending order by rearranging the terms in descending order of their exponents.

Therefore, the factored form of the expression 125x^3 + 343y^3 is:

(5x + 7y)(25x^2 - 35xy + 49y^2)

Conclusion

In this article, we have factored the expression 125x^3 + 343y^3 using the sum of cubes formula. We have expressed each factor as a polynomial in descending order and provided a clear understanding of the concept of factoring. This article is a valuable resource for students and teachers who want to learn more about factoring and algebra.

Example Use Cases

Factoring is a fundamental concept in algebra that has numerous applications in various fields, including:

  • Mathematics: Factoring is used to solve equations and inequalities, and to find the roots of polynomials.
  • Physics: Factoring is used to solve problems involving motion, energy, and momentum.
  • Engineering: Factoring is used to design and analyze systems, including electrical, mechanical, and civil engineering systems.

Tips and Tricks

Here are some tips and tricks to help you factor expressions:

  • Use the sum of cubes formula: The sum of cubes formula is a powerful tool for factoring expressions of the form a^3 + b^3.
  • Look for common factors: Look for common factors in the expression, such as a factor of 2 or a factor of 3.
  • Use the distributive property: The distributive property states that a(b + c) = ab + ac. Use this property to factor expressions.

Common Mistakes to Avoid

Here are some common mistakes to avoid when factoring expressions:

  • Not using the sum of cubes formula: The sum of cubes formula is a powerful tool for factoring expressions of the form a^3 + b^3. Make sure to use it when factoring expressions of this form.
  • Not looking for common factors: Look for common factors in the expression, such as a factor of 2 or a factor of 3.
  • Not using the distributive property: The distributive property states that a(b + c) = ab + ac. Use this property to factor expressions.

Conclusion

Introduction

In our previous article, we factored the expression 125x^3 + 343y^3 using the sum of cubes formula. In this article, we will provide a Q&A section to help you better understand the concept of factoring and its applications.

Q&A

Q: What is the sum of cubes formula?

A: The sum of cubes formula is a mathematical formula that states:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

This formula can be used to factor expressions of the form a^3 + b^3.

Q: How do I use the sum of cubes formula to factor an expression?

A: To use the sum of cubes formula to factor an expression, follow these steps:

  1. Check if the expression is in the form a^3 + b^3.
  2. If it is, use the sum of cubes formula to factor the expression.
  3. Simplify the expression to get the final factored form.

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

A: Here are some common mistakes to avoid when factoring expressions:

  • Not using the sum of cubes formula: The sum of cubes formula is a powerful tool for factoring expressions of the form a^3 + b^3. Make sure to use it when factoring expressions of this form.
  • Not looking for common factors: Look for common factors in the expression, such as a factor of 2 or a factor of 3.
  • Not using the distributive property: The distributive property states that a(b + c) = ab + ac. Use this property to factor expressions.

Q: How do I express each factor as a polynomial in descending order?

A: To express each factor as a polynomial in descending order, follow these steps:

  1. Look at each factor and identify the terms with the highest exponent.
  2. Arrange the terms in descending order of their exponents.
  3. Simplify the expression to get the final factored form.

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

A: Factoring has numerous applications in various fields, including:

  • Mathematics: Factoring is used to solve equations and inequalities, and to find the roots of polynomials.
  • Physics: Factoring is used to solve problems involving motion, energy, and momentum.
  • Engineering: Factoring is used to design and analyze systems, including electrical, mechanical, and civil engineering systems.

Q: How do I use factoring to solve problems in mathematics?

A: To use factoring to solve problems in mathematics, follow these steps:

  1. Identify the type of problem you are trying to solve.
  2. Use factoring to simplify the expression and solve the problem.
  3. Check your answer to make sure it is correct.

Q: What are some tips and tricks for factoring expressions?

A: Here are some tips and tricks for factoring expressions:

  • Use the sum of cubes formula: The sum of cubes formula is a powerful tool for factoring expressions of the form a^3 + b^3.
  • Look for common factors: Look for common factors in the expression, such as a factor of 2 or a factor of 3.
  • Use the distributive property: The distributive property states that a(b + c) = ab + ac. Use this property to factor expressions.

Conclusion

In conclusion, factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we have provided a Q&A section to help you better understand the concept of factoring and its applications. We have also provided tips and tricks for factoring expressions and common mistakes to avoid.