To factor the expression \( 40x^{4} - 5xy^{6} \), follow these steps:

1. **Factor Out the Greatest Common Factor (GCF):**
   
   Both terms have a common factor of \( 5x \):
   
   \[
   40x^{4} - 5xy^{6} = 5x(8x^{3} - y^{6})
   \]

2. **Recognize a Difference of Cubes:**
   
   Notice that \( 8x^{3} \) is \( (2x)^{3} \) and \( y^{6} \) is \( (y^{2})^{3} \). Thus, the expression inside the parentheses is a difference of cubes:
   
   \[
   8x^{3} - y^{6} = (2x)^{3} - (y^{2})^{3}
   \]

3. **Apply the Difference of Cubes Formula:**
   
   The difference of cubes formula is:
   
   \[
   a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
   \]
   
   Applying this to \( (2x)^{3} - (y^{2})^{3} \):
   
   \[
   (2x)^{3} - (y^{2})^{3} = (2x - y^{2})(4x^{2} + 2xy^{2} + y^{4})
   \]
   
4. **Combine the Factors:**
   
   Putting it all together:
   
   \[
   40x^{4} - 5xy^{6} = 5x(2x - y^{2})(4x^{2} + 2xy^{2} + y^{4})
   \]

**Final Factored Form:**

\[
5x \left( 2x - y^{2} \right) \left( 4x^{2} + 2xy^{2} + y^{4} \right)
\]

Home Question Bank Math Algebra

Question

2. \( 40 x^{4}-5 x y^{6} \)

Ask by Edwards Thornton. in South Africa

Jan 26,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The expression \( 40x^{4} - 5xy^{6} \) can be factored as \( 5x(2x - y^{2})(4x^{2} + 2xy^{2} + y^{4}) \).

Solution

To factor the expression \( 40x^{4} - 5xy^{6} \), follow these steps: 1. **Factor Out the Greatest Common Factor (GCF):** Both terms have a common factor of \( 5x \): \[ 40x^{4} - 5xy^{6} = 5x(8x^{3} - y^{6}) \] 2. **Recognize a Difference of Cubes:** Notice that \( 8x^{3} \) is \( (2x)^{3} \) and \( y^{6} \) is \( (y^{2})^{3} \). Thus, the expression inside the parentheses is a difference of cubes: \[ 8x^{3} - y^{6} = (2x)^{3} - (y^{2})^{3} \] 3. **Apply the Difference of Cubes Formula:** The difference of cubes formula is: \[ a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}) \] Applying this to \( (2x)^{3} - (y^{2})^{3} \): \[ (2x)^{3} - (y^{2})^{3} = (2x - y^{2})(4x^{2} + 2xy^{2} + y^{4}) \] 4. **Combine the Factors:** Putting it all together: \[ 40x^{4} - 5xy^{6} = 5x(2x - y^{2})(4x^{2} + 2xy^{2} + y^{4}) \] **Final Factored Form:** \[ 5x \left( 2x - y^{2} \right) \left( 4x^{2} + 2xy^{2} + y^{4} \right) \]

Reviewed and approved by the UpStudy tutoring team

error msg

Explain

Simplify this solution

Beyond the Answer

Did you know that polynomials like \( 40 x^{4}-5 x y^{6} \) can model a variety of real-world phenomena, from physics to economics? For example, in physics, polynomial expressions can represent trajectories of projectiles or the area under curves, all of which are influenced by certain factors that can be represented by variables like \( x \) and \( y \). When manipulating polynomials, a common mistake is forgetting to consider all terms when factoring or simplifying. Always look for a common factor in each term, like the 5 from your expression! Reducing it can make your work much easier, such as simplifying it to \( 5(8 x^{4}-y^{6}) \). This not only aids in clearer calculations but also in understanding the underlying relationships in the problem.

Latest Algebra Questions

Algebra United States Jan 28, 2025

Prev Next