Certainly! Let's factor the expression [tex]\(81 x^4 - 16\)[/tex] completely, step by step.
### Step 1: Recognize the Structure
Notice that [tex]\(81 x^4 - 16\)[/tex] resembles a difference of squares form: [tex]\(a^2 - b^2\)[/tex], where:
[tex]\[ a^2 = (9x^2)^2 \quad \text{and} \quad b^2 = 4^2 \][/tex]
Thus, we can rewrite the expression as:
[tex]\[ (9x^2)^2 - 4^2 \][/tex]
### Step 2: Apply the Difference of Squares Formula
Recall the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Applying this to our expression:
[tex]\[ (9x^2)^2 - 4^2 = (9x^2 - 4)(9x^2 + 4) \][/tex]
### Step 3: Factor Further if Possible
Next, we look at each factor to determine if they can be factored further.
#### Factor [tex]\(9x^2 - 4\)[/tex]
This again is a difference of squares and can be factored further. Write:
[tex]\[ 9x^2 - 4 = (3x)^2 - 2^2 \][/tex]
Again, apply the difference of squares formula:
[tex]\[ (3x)^2 - 2^2 = (3x - 2)(3x + 2) \][/tex]
#### Factor [tex]\(9x^2 + 4\)[/tex]
The term [tex]\(9x^2 + 4\)[/tex] is a sum of squares and cannot be factored further over the real numbers. Hence, it remains as it is.
### Step 4: Combine All Factors
Combine all the factors we obtained:
[tex]\[ 81 x^4 - 16 = (3 x - 2)(3 x + 2) (9 x^2 + 4) \][/tex]
### Conclusion
Thus, the completely factored form of [tex]\(81 x^4 - 16\)[/tex] is:
[tex]\[
(3 x - 2)(3 x + 2)(9 x^2 + 4)
\][/tex]
