Question

Utilizing the polynomial identities theorem, how can you factor the expression 64x^3 - 1?

a) (4x - 1)(16x^2 + 4x + 1)
b) (4x + 1)(16x^2 - 4x + 1)
c) (2x - 1)(8x^2 + 2x + 1)
d) (2x + 1)(8x^2 - 2x + 1)

Answer 1

To factor the expression 64x^3 - 1 using the polynomial identities theorem, we can utilize the difference of cubes identity. The correct factorization is (4x - 1)(16x^2 + 4x + 1).

Step-by-step explanation:

To factor the expression 64x^3 - 1 using the polynomial identities theorem, we can utilize the difference of cubes identity. The difference of cubes identity states that a^3 - b^3 = (a - b)(a^2 +ab + b^2). In this case, we have 64x^3 - 1, which can be expressed as (4x)^3 - 1^3. Applying the difference of cubes identity, we get (4x - 1)((4x)^2 + (4x)(1) + 1^2). Simplifying further, we have (4x - 1)(16x^2 + 4x + 1). Therefore, the correct factorization is option a) (4x - 1)(16x^2 + 4x + 1).