Final answer:
Both Statement 1 and Statement 2 are incorrect representations of the polynomial identity for x^3 + 64. Neither simplifies correctly to the given expression, so both are invalid polynomial identities. The correct answer is option: d) Statement 1 and statement 2 are invalid.
Step-by-step explanation:
We need to determine which statement accurately represents the polynomial identity for x³+ 64. This can be fact-checked by using polynomial identities or performing polynomial division.
Statement 1 suggests that x³ + 64 can be factored as (x + 4)(x² + 4x + 16). Let's check this by expanding the right-hand side:
- (x + 4)(x² + 4x + 16) = x³ + 4x² + 16x + 4x² + 16x + 64
- = x³ + 8x² + 32x + 64
Upon expanding, we see that this doesn't equal x³ + 64, but rather x³ + 8x² + 32x + 64. Therefore, Statement 1 is not a valid polynomial identity.
Statement 2 suggests that x³ + 64 equals (x - 4)(x²+ 4x + 16). This is known as the sum of cubes identity.
The correct sum of cubes identity is ( x³ + a³) = (x + a)(x²- ax + a²) when a = 4.
- (x - 4)(x² + 4x + 16) = x³ - 4x² + 16x - 4x² - 16x + 64
- = x³ - 8x² + 64
Upon expanding, it's clear that this does not equal x³ + 64. Therefore, Statement 2 is not a valid polynomial identity either.
Both statements are invalid polynomial identities, as they do not simplify correctly to x³ + 64.