To factor the expression 32x³ + 4, we start by looking for common factors in both terms. Noticing that both terms share a factor of 4, we can factor that out:
32x³ + 4 = 4(8x³ + 1)
Next, we recognize that 8x³ + 1 can be expressed as a sum of cubes, since 8x³ is (2x)³ and 1 is 1³. The sum of cubes can be factored using the formula a³ + b³ = (a + b)(a² – ab + b²).
In our case, let a = 2x and b = 1. Applying the sum of cubes formula, we have:
- a + b = 2x + 1
- a² = (2x)² = 4x²
- ab = (2x)(1) = 2x
- b² = 1² = 1
Now substituting these into the formula gives:
(2x + 1)(4x² – 2x + 1)
Finally, putting it all together, the fully factored form of 32x³ + 4 is:
32x³ + 4 = 4(2x + 1)(4x² – 2x + 1)