Factorize completely 192-3X^6

Question

 This question is very tricky to me I want you to factorize completely the question it is also a pure Mathematics unit 1 question 

Answer 1

192 - 3x⁶ = 3[64 - x⁶]

 

               = 3[4³ - (x²)³]

 

               = 3[(4 - x²)(4² + 4x² + (x²)²)]

 

               = 3(2-x)(2+x)(x⁴ + 4x² + 16)

 

Answer 2

If x = 1, the ORIGINAL expression has the value 189, but Andrew's answer, evaluated at x=1, is 3(1)(3)(3)(3) and so this counter example reveals an error.  Mark's answer checks.

Answer 3

192 - 3x⁶

= 3(64 - x⁶)

= 3(4³ - (x²)³)

 

Difference of two cubes:

(a³ - b³) = (a-b)(a²+ab+b²), a=4, b=x²

 

3[(4-x²)(4²+4x²+x⁴)]

 

For (4-x²) use (a²-b²) = (a-b)(a+b)

4-x² = (2-x)(2+x)

 

3(2-x)(2+x)(x⁴+4x²+16)

 

Look at (x⁴+4x²+16)

Let u=x²

u²+4u+16

Since there are no factors of 16 that will add to 4, this

is not really factorable. Thus the final answer is:

3(2-x)(2+x)(x⁴ + 4x² + 16)