Write a linear factorization of the function. f(x) = x4 + 36x2

Solution:

Given function f(x) = x4 + 36x2

= (x2)2 + (6x)2

We know that a2 + b2 = (a + b)2 - 2ab

Therefore,

= (x2)2 + (6x)2

= (x2 + 6x)2 - 2(x2)(6x)

= x2(x - 6)2 - 12x3

= x2[(x - 6)2 - 12x]

= x2[(x - 6)2 - (√12x)2 ]

We know that a2 - b2 = (a + b)(a - b)

(x - 6)2 - (√12x)2 = (x - 6 + √12x)(x - 6 - √12x), therefore

x2[(x - 6)2 - (√12x)2 ]

= x2(x - 6 + √12x)(x - 6 - √12x)

Alternatively

f(x) = x4 + 36x2

= x2(x2 +36)

= x2(x + 6i)(x - 6i)

The linear factors are x, (x - 6i) and (x + 6i)

Write a linear factorization of the function. f(x) = x4 + 36x2

Summary:

Linear factorization of the function. f(x) = x4 + 36x2 leads to x2(x + 6i)(x - 6i)

Math worksheets and
visual curriculum