To analyze the expression ( 2x^4 - 16x^2 + 32 ), we can start by factoring it.
First, let's factor out the common term:
[
2(x^4 - 8x^2 + 16)
]
Now, we can simplify the quadratic part ( x^4 - 8x^2 + 16 ) by letting ( y = x^2 ), which transforms our expression into a quadratic form:
[
y^2 - 8y + 16
]
Next, we can factor the quadratic:
[
(y - 4)^2
]
Substituting back ( y = x^2 ), we have:
[
(x^2 - 4)^2
]
Now, we can further factor ( x^2 - 4 ) as it is a difference of squares:
[
(x - 2)(x + 2)
]
Therefore, we substitute back to get the complete factorization of the original expression:
[
2[(x - 2)(x + 2)]^2
]
Putting it all together, the fully factored form of ( 2x^4 - 16x^2 + 32 ) is:
[
2(x - 2)^2(x + 2)^2
]
This expression can be used in further calculations or to find roots if needed. If you have any specific questions or need further explanations, let me know!
