The linear factorization of the function is f(x) = x² (x + 3i) (x - 3i) (option A)
Step-by-step explanation:
f(x)=x^4+9x^2
we factorize x² as it is common to both
f(x) = x²(x² + 9)
The next thing is to factorize (x² + 9)
To do that we need to introduce complex numbers as it is not possible to factorize sum of two squares with real numbers.
(x² + 9) = x² + 3² (sum of two squares)
the complex number: i² = -1
x² + 3² = x² +(1) 3² = x² - (-1)(3²)
x² - (-1)(3²) = x² - (i²)(3²)
recall difference of two squares:
a² - b² = (a+b) (a-b)
x² - (i²)(3²) = x² - (3i²)
= (x + 3i) (x - 3i)
(x² + 9) = (x + 3i) (x - 3i)
f(x) = x²(x² + 9) = x² (x + 3i) (x - 3i)
The linear factorization of the function is f(x) = x² (x + 3i) (x - 3i) (option A)