(2k – 1)(4k2 + 2k + 1)

Jasonbradley

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Jun 16, 2024

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Factoring the Expression (2k – 1)(4k² + 2k + 1)

This expression represents a special case of factoring known as the difference of cubes. Let's break it down:

Understanding the Difference of Cubes

The difference of cubes pattern states: **a³ - b³ = (a - b)(a² + ab + b²) **

Applying the Pattern

In our expression, we can see:

• a³ = (2k)³ = 8k³
• b³ = (1)³ = 1

Therefore, we can rewrite the expression as:

(2k)³ - (1)³

Now, directly applying the difference of cubes pattern:

(2k - 1)((2k)² + (2k)(1) + (1)²)

Simplifying further:

(2k - 1)(4k² + 2k + 1)

Conclusion

The fully factored form of the expression (2k – 1)(4k² + 2k + 1) is itself. It's already in its simplest factored form, representing the difference of cubes.