Let's go through each expression step-by-step to factorize them properly.
### Expression 1: [tex]\( 3 x^2 - 12 y^2 \)[/tex]
1. First, we factor out the common factor of 3:
[tex]\[ 3 x^2 - 12 y^2 = 3(x^2 - 4 y^2) \][/tex]
2. Next, we recognize that [tex]\( x^2 - 4 y^2 \)[/tex] is a difference of squares:
[tex]\[ x^2 - 4 y^2 = (x + 2y)(x - 2y) \][/tex]
3. Substituting this back into our expression, we get:
[tex]\[ 3(x^2 - 4 y^2) = 3(x + 2y)(x - 2y) \][/tex]
So, the final factors are:
[tex]\[ \boxed{3(x + 2y)(x - 2y)} \][/tex]
### Expression 2: [tex]\( 1 - 16 x^8 \)[/tex]
1. Recognize that [tex]\( 1 - 16 x^8 \)[/tex] is a difference of squares:
[tex]\[ 1 - (4x^4)^2 \][/tex]
2. Factor the difference of squares:
[tex]\[ 1 - (4x^4)^2 = (1 + 4x^4)(1 - 4x^4) \][/tex]
3. Next, factor [tex]\( 1 - 4x^4 \)[/tex] again because it can be further factored as another difference of squares:
[tex]\[ 1 - 4x^4 = (1 - (2x^2)^2) = (1 + 2x^2)(1 - 2x^2) \][/tex]
4. So we combine all factors:
[tex]\[ 1 - 16 x^8 = (1 + 4x^4)(1 + 2x^2)(1 - 2x^2) \][/tex]
So, the final factors are:
[tex]\[ \boxed{(1 + 4x^4)(1 + 2x^2)(1 - 2x^2)} \][/tex]
### Expression 3: [tex]\( a^4 - 625 b^8 \)[/tex]
1. Recognize that [tex]\( a^4 - 625 b^8 \)[/tex] is a difference of squares:
[tex]\[ (a^2)^2 - (25 b^4)^2 \][/tex]
2. Factor this as a difference of squares:
[tex]\[ (a^2)^2 - (25 b^4)^2 = (a^2 + 25 b^4)(a^2 - 25 b^4) \][/tex]
3. Next, factor [tex]\( a^2 - 25 b^4 \)[/tex] again because it can be further factored as another difference of squares:
[tex]\[ a^2 - 25 b^4 = (a - 5 b^2)(a + 5 b^2) \][/tex]
4. So we combine all factors:
[tex]\[ a^4 - 625 b^8 = (a^2 + 25 b^4)(a - 5 b^2)(a + 5 b^2) \][/tex]
So, the final factors are:
[tex]\[ \boxed{(a^2 + 25 b^4)(a - 5 b^2)(a + 5 b^2)} \][/tex]
To summarize:
### Final Factors:
1. [tex]\( 3(x + 2y)(x - 2y) \)[/tex]
2. [tex]\( (1 + 4x^4)(1 + 2x^2)(1 - 2x^2) \)[/tex]
3. [tex]\( (a^2 + 25 b^4)(a - 5 b^2)(a + 5 b^2) \)[/tex]
