To factor the expression [tex]\( 81x^4 - y^4 \)[/tex] completely, let's follow through the factoring process step by step, using algebraic techniques.
1. Identify the expression as a difference of squares:
We start with:
[tex]\[
81x^4 - y^4
\][/tex]
Notice that both [tex]\( 81x^4 \)[/tex] and [tex]\( y^4 \)[/tex] are perfect squares. This allows us to write the expression as a difference of squares:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
where [tex]\( a^2 = 81x^4 \)[/tex] and [tex]\( b^2 = y^4 \)[/tex]. This gives us:
[tex]\[
a = 9x^2 \quad \text{and} \quad b = y^2
\][/tex]
2. Factor the difference of squares:
Applying the difference of squares formula to [tex]\( 81x^4 - y^4 \)[/tex], we get:
[tex]\[
81x^4 - y^4 = (9x^2)^2 - (y^2)^2 = (9x^2 - y^2)(9x^2 + y^2)
\][/tex]
3. Further factor the [tex]\( 9x^2 - y^2 \)[/tex] term:
Now, we need to factor [tex]\( 9x^2 - y^2 \)[/tex], which is also a difference of squares:
[tex]\[
9x^2 - y^2 = (3x)^2 - (y)^2 = (3x - y)(3x + y)
\][/tex]
4. Combine all factors:
We now have factored [tex]\( 81x^4 - y^4 \)[/tex] as:
[tex]\[
81x^4 - y^4 = (9x^2 + y^2)(3x - y)(3x + y)
\][/tex]
So, the completely factored form of [tex]\( 81x^4 - y^4 \)[/tex] is:
[tex]\[
(9x^2 + y^2)(3x - y)(3x + y)
\][/tex]
Therefore, the correct answer is:
A) [tex]\(\left(9 x^2 + y^2\right) (3 x + y) (3 x - y)\)[/tex]
