Sure! Let’s factorize each of the given expressions step-by-step.
### a) Factorize [tex]\(x^4 - 16\)[/tex]
1. Notice that [tex]\(16\)[/tex] is [tex]\(2^4\)[/tex], so we can rewrite the expression as:
[tex]\[
x^4 - 16 = x^4 - 2^4
\][/tex]
2. This is a difference of squares:
[tex]\(
(x^2 - 2^2)(x^2 + 2^2)
\)[/tex]
Which simplifies to:
[tex]\[
(x^2 - 4)(x^2 + 4)
\][/tex]
3. The term [tex]\(x^2 - 4\)[/tex] can be further factorized as:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
4. Thus,
[tex]\[
x^4 - 16 = (x - 2)(x + 2)(x^2 + 4)
\][/tex]
### b) Factorize [tex]\(a^4 - 81\)[/tex]
1. Notice that [tex]\(81\)[/tex] is [tex]\(3^4\)[/tex], so we can rewrite the expression as:
[tex]\[
a^4 - 81 = a^4 - 3^4
\][/tex]
2. This is a difference of squares:
[tex]\(
(a^2 - 3^2)(a^2 + 3^2)
\)[/tex]
Which simplifies to:
[tex]\[
(a^2 - 9)(a^2 + 9)
\][/tex]
3. The term [tex]\(a^2 - 9\)[/tex] can be further factorized as:
[tex]\[
a^2 - 9 = (a - 3)(a + 3)
\][/tex]
4. Thus,
[tex]\[
a^4 - 81 = (a - 3)(a + 3)(a^2 + 9)
\][/tex]
### c) [tex]\(x^4 -\)[/tex]
- The expression provided is incomplete, making it impossible to factorize.
### e) Factorize [tex]\(81 x^4 - 625\)[/tex]
1. Notice that [tex]\(81\)[/tex] is [tex]\(9^2\)[/tex] and [tex]\(625\)[/tex] is [tex]\(25^2\)[/tex], so we can rewrite the expression as:
[tex]\[
81 x^4 - 625 = (9x^2)^2 - 25^2
\][/tex]
2. This is a difference of squares:
[tex]\[
(9x^2 - 25)(9x^2 + 25)
\][/tex]
3. The term [tex]\(9x^2 - 25\)[/tex] can be further factorized as:
[tex]\[
9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)
\][/tex]
4. Thus,
[tex]\[
81 x^4 - 625 = (3x - 5)(3x + 5)(9x^2 + 25)
\][/tex]
### f) Factorize [tex]\(32 y^4 - 162\)[/tex]
1. Factor out the common factor [tex]\(2\)[/tex]:
[tex]\[
32 y^4 - 162 = 2(16 y^4 - 81)
\][/tex]
2. Notice that [tex]\(16\)[/tex] is [tex]\(4^2\)[/tex] and [tex]\(81\)[/tex] is [tex]\(9^2\)[/tex], so we can rewrite the expression inside the parentheses as:
[tex]\[
16 y^4 - 81 = (4y^2)^2 - 9^2
\][/tex]
3. This is a difference of squares:
[tex]\(
(4y^2 - 9)(4y^2 + 9)
\)[/tex]
4. The term [tex]\(4y^2 - 9\)[/tex] can be further factorized as:
[tex]\[
4y^2 - 9 = (2y)^2 - 3^2 = (2y - 3)(2y + 3)
\][/tex]
5. Thus,
[tex]\[
32 y^4 - 162 = 2(2y - 3)(2y + 3)(4y^2 + 9)
\][/tex]
### g) [tex]\(a^8 -\)[/tex]
- The expression provided is incomplete, making it impossible to factorize.
In summary, the factorizations are:
a) [tex]\(x^4 - 16 = (x - 2)(x + 2)(x^2 + 4)\)[/tex]
b) [tex]\(a^4 - 81 = (a - 3)(a + 3)(a^2 + 9)\)[/tex]
c) Incomplete, cannot be factorized.
e) [tex]\(81 x^4 - 625 = (3x - 5)(3x + 5)(9x^2 + 25)\)[/tex]
f) [tex]\(32 y^4 - 162 = 2(2y - 3)(2y + 3)(4y^2 + 9)\)[/tex]
g) Incomplete, cannot be factorized.
