To factor the polynomial expression [tex]\( x^4 + 18x^2 + 81 \)[/tex], let's break it down step-by-step.
### Step 1: Recognize the Structure
Notice that the polynomial [tex]\( x^4 + 18x^2 + 81 \)[/tex] can be seen as a quadratic in terms of [tex]\( x^2 \)[/tex]:
[tex]\[ (x^2)^2 + 18(x^2) + 81 \][/tex]
### Step 2: Compare with Perfect Square Form
We can compare this expression to the expanded form of a perfect square trinomial:
[tex]\[ (a^2 + 2ab + b^2) = (a + b)^2 \][/tex]
We are looking for a form like [tex]\( (x^2 + a)^2 \)[/tex]. Let's expand [tex]\( (x^2 + a)^2 \)[/tex]:
[tex]\[ (x^2 + a)^2 = (x^2)^2 + 2(x^2)a + a^2 \][/tex]
[tex]\[ = x^4 + 2ax^2 + a^2 \][/tex]
### Step 3: Identify [tex]\( a \)[/tex]
To match the structure of [tex]\( x^4 + 18x^2 + 81 \)[/tex] with [tex]\( x^4 + 2ax^2 + a^2 \)[/tex], we compare coefficients:
- [tex]\( 2a = 18 \)[/tex]
- [tex]\( a^2 = 81 \)[/tex]
Solving for [tex]\( a \)[/tex] from [tex]\( 2a = 18 \)[/tex], we get:
[tex]\[ 2a = 18 \][/tex]
[tex]\[ a = 9 \][/tex]
And indeed:
[tex]\[ (9)^2 = 81 \][/tex]
### Step 4: Factor the Expression
With [tex]\( a = 9 \)[/tex], we can rewrite the polynomial:
[tex]\[ x^4 + 18x^2 + 81 = (x^2 + 9)^2 \][/tex]
Thus, the factorization of [tex]\( x^4 + 18x^2 + 81 \)[/tex] is:
[tex]\[ (x^2 + 9)^2 \][/tex]
### Final Factored Form
Therefore, the factored form of the given polynomial is [tex]\( (x^2 + 9)^2 \)[/tex].
So in terms of the missing expressions for the given format, you would have:
```
[tex]\(\left.x^4+18x^2+81=(x^2+9)^2\right.\)[/tex]
```
