Write a linear factorization of the function. f(x) = x⁴ + 16x²

Solution:

Given function is

f(x) = x⁴ + 16x²

A factorization of a polynomial in which every factor obtained is a linear factor is said to be linear factorization.

Factoring the polynomial x⁴ + 16x² by taking x² as common from the terms.

f(x) = x²{x² +16}

f(x) = x²{x² - (-16)}

f(x) = x²{x² - (16i²)}

f(x) = x²{x² - (4i)²}

This resembles a² - b² = (a + b)(a - b)

f(x) = x²{(x + 4i)(x - 4i)}

Therefore, f(x) is linearly factored as x²{(x + 4i)(x - 4i)}

Write a linear factorization of the function. f(x) = x⁴ + 16x²

Summary:

A linear factorization of the function. f(x) = x⁴ + 16x² is x²{(x + 4i)(x - 4i)}.