Question

Every polynomial with complex coefficients can be written as the product of linear factors. Enter the linear factors of P(z) = z⁴ – 256 separated by commas. For example you could enter three linear factors as: (z – 4), z + 3 + 2i, z

Answer 1

The linear factors of P(z) = z⁴ – 256 are (z + 4i)(z - 4i)(z + 4)(z - 4).

Step-by-step explanation:

Every polynomial with complex coefficients can be factored completely into linear factors. To find the linear factors of the polynomial P(z) = z⁴ – 256, we need to factorize the difference of squares, which is the form a² – b². The difference of squares factors into (a + b)(a - b). So, applying this to our polynomial, we have:

P(z) = z⁴ – 256

P(z) = (z² + 16)(z² - 16)

The linear factors of P(z) = z⁴ – 256 are (z + 4i)(z - 4i)(z + 4)(z - 4).