Question

Which of the following statements is a valid polynomial identity? Statement 1: x3 + 64 = (x + 4)(x2 + 4x + 16) Statement 2: x3 + 64 = (x − 4)(x2 + 4x + 16) Group of answer choices Only statement 1 is valid. Only statement 2 is valid. Statement 1 and statement 2 are valid. Statement 1 and statement 2 are invalid.

Answer 1

Both statements provided are invalid as expanding the right-hand side of each equation does not result in an expression that is equal to the left-hand side, which is x³+ 64. The correct answer is option: Statement 1 and statement 2 are invalid.

Step-by-step explanation:

The question tests the knowledge of polynomial identities, specifically determining the validity of two given statements regarding the expression x³ + 64.

We can verify each statement by expanding the right side of the equation and comparing it to the left side.

For Statement 1: x³ + 64 = (x + 4)(x² - 4x + 16).

To test this, we need to expand the right-hand side:

• (x + 4)(x² + 4x + 16) = x³ + 4x²+ 16x + 4x² + 16x + 64
• Combining like terms gives us x³ + 8x² + 32x + 64

This is not equal to x³ + 64, so Statement 1 is invalid.

For Statement 2: x³ + 64 = (x - 4)(x² + 4x + 16). Expanding the right-hand side gives:

• (x - 4)(x² + 4x + 16)
• = x³ - 4x² + 16x - 4x² + 16x - 64
• Combining like terms results in x³- 8x² + 32x - 64

This also does not equal x³ + 64, so Statement 2 is also invalid.

Therefore, the correct answer is that both Statement 1 and Statement 2 are invalid.