Which Of The Following Is The Correct Factorization Of The Polynomial Below?$\[ P^3 - 343q^3 \\]A. \[$(p - 49q)(p^2 + 7pq + 49q^2)\$\]B. \[$(p - 7q)(p^2 + 7pq + 49q^2)\$\]C. \[$(p^2 + 7q)(p^3 + 49pq + 7q^2)\$\]D. The

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Introduction

In mathematics, factorization is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on the factorization of a given polynomial, which is a crucial aspect of algebra. We will explore the different options provided and determine the correct factorization of the polynomial.

Understanding the Polynomial

The given polynomial is p3−343q3p^3 - 343q^3. To factorize this polynomial, we need to identify its roots and express it as a product of linear factors. The polynomial can be rewritten as:

p3−73q3p^3 - 7^3q^3

Applying the Difference of Cubes Formula

The difference of cubes formula states that:

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

We can apply this formula to the given polynomial by substituting a=pa = p and b=7qb = 7q. This gives us:

(p−7q)(p2+7pq+49q2)(p - 7q)(p^2 + 7pq + 49q^2)

Evaluating the Options

Now that we have the correct factorization of the polynomial, let's evaluate the options provided:

Option A

(p−49q)(p2+7pq+49q2)(p - 49q)(p^2 + 7pq + 49q^2)

This option is incorrect because the first term in the factorization is p−49qp - 49q, which is not equal to p−7qp - 7q.

Option B

(p−7q)(p2+7pq+49q2)(p - 7q)(p^2 + 7pq + 49q^2)

This option is correct because it matches the factorization we obtained using the difference of cubes formula.

Option C

(p2+7q)(p3+49pq+7q2)(p^2 + 7q)(p^3 + 49pq + 7q^2)

This option is incorrect because the first term in the factorization is p2+7qp^2 + 7q, which is not equal to p2+7pqp^2 + 7pq.

Conclusion

In conclusion, the correct factorization of the polynomial p3−343q3p^3 - 343q^3 is:

(p−7q)(p2+7pq+49q2)(p - 7q)(p^2 + 7pq + 49q^2)

This factorization can be obtained using the difference of cubes formula, which is a fundamental concept in algebra. By applying this formula, we can express the polynomial as a product of simpler expressions, making it easier to analyze and solve.

Final Answer

The final answer is:

  • Option B: (p−7q)(p2+7pq+49q2)(p - 7q)(p^2 + 7pq + 49q^2)

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Introduction

In our previous article, we explored the factorization of the polynomial p3−343q3p^3 - 343q^3. We applied the difference of cubes formula to obtain the correct factorization, which is:

(p−7q)(p2+7pq+49q2)(p - 7q)(p^2 + 7pq + 49q^2)

In this article, we will provide a Q&A guide to help you understand the concept of factorization and how to apply it to different types of polynomials.

Q&A: Factorization of the Polynomial

Q: What is factorization?

A: Factorization is the process of expressing an algebraic expression as a product of simpler expressions.

Q: What is the difference of cubes formula?

A: The difference of cubes formula states that:

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Q: How do I apply the difference of cubes formula?

A: To apply the difference of cubes formula, you need to identify the values of aa and bb in the given polynomial. Then, substitute these values into the formula and simplify.

Q: What are some common types of polynomials that can be factorized using the difference of cubes formula?

A: Some common types of polynomials that can be factorized using the difference of cubes formula include:

  • a3−b3a^3 - b^3
  • a3+b3a^3 + b^3
  • a3−2ab2a^3 - 2ab^2
  • a3+2ab2a^3 + 2ab^2

Q: How do I determine the correct factorization of a polynomial?

A: To determine the correct factorization of a polynomial, you need to apply the difference of cubes formula and simplify the expression. You can also use other factorization techniques, such as factoring out common factors or using the quadratic formula.

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

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

  • Not identifying the correct values of aa and bb in the difference of cubes formula
  • Not simplifying the expression correctly
  • Not factoring out common factors
  • Not using the quadratic formula when necessary

Conclusion

In conclusion, factorization is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. By applying the difference of cubes formula and other factorization techniques, you can determine the correct factorization of a polynomial. Remember to avoid common mistakes and simplify the expression correctly to obtain the correct factorization.

Final Tips

  • Practice factorizing different types of polynomials to become more comfortable with the concept.
  • Use the difference of cubes formula and other factorization techniques to simplify expressions.
  • Avoid common mistakes and double-check your work to ensure accuracy.

Common Factorization Formulas

Here are some common factorization formulas that you can use to simplify expressions:

  • Difference of Squares Formula: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
  • Sum of Squares Formula: a2+b2=(a+b)2−2aba^2 + b^2 = (a + b)^2 - 2ab
  • Difference of Cubes Formula: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
  • Sum of Cubes Formula: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of factorization and how to apply it to different types of polynomials. We also covered common factorization formulas and provided final tips to help you become more comfortable with the concept. Remember to practice factorizing different types of polynomials and to use the difference of cubes formula and other factorization techniques to simplify expressions.