Sure, let's factorize each expression step-by-step.
### Question 1
1. Expression to factorize:
[tex]\[
3x^2 - 12y^2
\][/tex]
2. Step-by-step factorization:
- Step 1: Factor out [tex]\(3\)[/tex] from the entire expression:
[tex]\[
3(x^2 - 4y^2)
\][/tex]
- Step 2: Recognize that [tex]\(x^2 - 4y^2\)[/tex] is a difference of squares. Recall that [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex]. Here [tex]\(a = x\)[/tex] and [tex]\(b = 2y\)[/tex]:
[tex]\[
x^2 - 4y^2 = (x + 2y)(x - 2y)
\][/tex]
- Step 3: Combine these factors to get the final expression:
[tex]\[
3(x + 2y)(x - 2y)
\][/tex]
Final factors:
[tex]\[
3(x + 2y)(x - 2y)
\][/tex]
### Question 2
1. Expression to factorize:
[tex]\[
1 - 16x^8
\][/tex]
2. Step-by-step factorization:
- Step 1: Recognize [tex]\(1 - 16x^8\)[/tex] as a difference of squares. Here [tex]\(a = 1\)[/tex] and [tex]\(b = (4x^4)\)[/tex]:
[tex]\[
1 - 16x^8 = (1 + 4x^4)(1 - 4x^4)
\][/tex]
- Step 2: Notice that [tex]\(1 - 4x^4\)[/tex] is itself a difference of squares. Continue factorizing:
[tex]\[
1 - 4x^4 = (1 + 2x^2)(1 - 2x^2)
\][/tex]
- Step 3: Substitute back:
[tex]\[
(1 + 4x^4)(1 + 2x^2)(1 - 2x^2)
\][/tex]
Final factors:
[tex]\[
(1 + 4x^4)(1 + 2x^2)(1 - 2x^2)
\][/tex]
### Question 3
1. Expression to factorize:
[tex]\[
a^4 - 625b^8
\][/tex]
2. Step-by-step factorization:
- Step 1: Recognize [tex]\(a^4 - 625b^8\)[/tex] as a difference of squares. Here [tex]\(a = a^2\)[/tex] and [tex]\(b = 25b^4\)[/tex]:
[tex]\[
a^4 - 625b^8 = (a^2 + 25b^4)(a^2 - 25b^4)
\][/tex]
- Step 2: Notice that [tex]\(a^2 - 25b^4\)[/tex] is also a difference of squares. Continue factorizing:
[tex]\[
a^2 - 25b^4 = (a + 5b^2)(a - 5b^2)
\][/tex]
- Step 3: Combine these factors:
[tex]\[
(a^2 + 25b^4)(a + 5b^2)(a - 5b^2)
\][/tex]
Final factors:
[tex]\[
(a^2 + 25b^4)(a + 5b^2)(a - 5b^2)
\][/tex]
By following these steps, we've successfully factorized the given expressions.
