Question

Write a linear factorization of the function.

f(x) = x4+ 9x2

f(x) = x2(x + 3i)(x - 3i)

f(x) = x2(x + 3i)2

f(x) = x2(3x + i)(3x - i)

f(x) = x2(3x + i)

Answer 1

`f(x)=x^2(x+3i)(x-3i)`

Explanation:

Given expression is,

`f(x)=x^4+9x^2`

`=x^2\{x^2+9\}` ( Converse of distributive property )

`=x^2\{x^2-i^2.9\}` ( i² = -1 )

`=x^2\{(x)^2-i^2(3)^2\}`

`=x^2\{(x)^2-(3i)^2\}`

`=x^2(x+3i)(x-3i)` ( a² - b² = (a+b)(a-b) )

Since, further factorization is not possible,

Thus, the required linear factorization of the function is,

`f(x)=x^2(x+3i)(x-3i)`

First option is correct.

Answer 2

f(x) = x^4 + 9x^2 = x^2(x^2 + 9) = x^2(x^2 - (-9)) = x^2(x - sqrt(-9))(x + sqrt(-9)) = x^2(x + 3i)(x - 3i)