Factorize 3x^4-48y^4
3(x^4-16y^4)
3(x^2-4y^2)(x^2+4y^2)
3(x-2y)(x+2y)(x^2+4y^2)
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To factorize the expression 3x^4 - 48y^4, we can first notice that both terms have a common factor of 3. Factoring out this common factor gives us:
3(x^4 - 16y^4)
Now, let's focus on the expression inside the parentheses: x^4 - 16y^4. This is a difference of squares because it can be written as (x^2)^2 - (4y^2)^2.
Using the formula for the difference of squares, we know that (a^2 - b^2) can be factored as (a + b)(a - b). Applying this to our expression:
x^4 - 16y^4 = (x^2 + 4y^2)(x^2 - 4y^2)
Now, let's further factor the expression x^2 - 4y^2. This is also a difference of squares because it can be written as (x)^2 - (2y)^2.
Applying the difference of squares formula once again:
x^2 - 4y^2 = (x + 2y)(x - 2y)
Therefore, the fully factorized expression is:
3(x^2 + 4y^2)(x + 2y)(x - 2y)
And that's how you factorize the expression 3x^4 - 48y^4.