https://file.aiquickdraw.com/mathbot/file/1726875002084tj9lfwej.jpg

Let's go through the table and factor each polynomial step by step to find the corresponding binomial factor:

1. **\( 343m^3 + 64n^3 \):**
   - This is a sum of cubes: \( a^3 + b^3 \), where \( a = 7m \) and \( b = 4n \).
   - The formula for factoring sums of cubes is:
     \[
     a^3 + b^3 = (a + b)(a^2 - ab + b^2)
     \]
   - So, \( 343m^3 + 64n^3 = (7m + 4n)((7m)^2 - (7m)(4n) + (4n)^2) \).
   - The binomial factor is **\( (7m + 4n) \)**.

2. **\( m^3 + 8n^3 \):**
   - This is again a sum of cubes: \( a^3 + b^3 \), where \( a = m \) and \( b = 2n \).
   - Using the same formula:
     \[
     m^3 + 8n^3 = (m + 2n)(m^2 - m(2n) + (2n)^2)
     \]
   - The binomial factor is **\( (m + 2n) \)**.

3. **\( 8x^4 + x \):**
   - Factor out the greatest common factor (GCF), which is \( x \):
     \[
     8x^4 + x = x(8x^3 + 1)
     \]
   - The binomial factor is **\( x \)**.

4. **\( x^3 - 216y^3 \):**
   - This is a difference of cubes: \( a^3 - b^3 \), where \( a = x \) and \( b = 6y \).
   - The formula for factoring differences of cubes is:
     \[
     a^3 - b^3 = (a - b)(a^2 + ab + b^2)
     \]
   - So, \( x^3 - 216y^3 = (x - 6y)(x^2 + x(6y) + (6y)^2) \).
   - The binomial factor is **\( (x - 6y) \)**.

5. **\( 64x^3 + 1 \):**
   - This is a sum of cubes: \( a^3 + b^3 \), where \( a = 4x \) and \( b = 1 \).
   - Using the sum of cubes formula:
     \[
     64x^3 + 1 = (4x + 1)((4x)^2 - (4x)(1) + 1^2)
     \]
   - The binomial factor is **\( (4x + 1) \)**.

### Summary of Binomial Factors:
1. \( 343m^3 + 64n^3 \) → \( (7m + 4n) \)
2. \( m^3 + 8n^3 \) → \( (m + 2n) \)
3. \( 8x^4 + x \) → \( x \)
4. \( x^3 - 216y^3 \) → \( (x - 6y) \)
5. \( 64x^3 + 1 \) → \( (4x + 1) \)

Would you like further details or explanations on any of these factorizations? Here are five related questions you could explore:

1. How do you factor a difference of squares?
2. Can you factor the sum of squares?
3. What is the general formula for factoring a cubic polynomial?
4. How do you determine the greatest common factor in a polynomial?
5. What are the differences between factoring a sum of cubes and a difference of cubes?

**Tip:** Always check if there's a common factor (GCF) before factoring more complex polynomials.

For each polynomial shown, select the binomial factor of the factored polynomial.

Learn how to factor various polynomials, including sums and differences of cubes. This detailed step-by-step solution covers key algebraic concepts and is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation