To solve the polynomial equation provided, let's break down the steps one-by-one to reach the solution.
Given the equation:
[tex]\[ x^4 + 18x^2 + 81 \][/tex]
First, observe that you can rewrite the polynomial in a way that will reveal its structure more clearly.
1. Identify the structure of the polynomial:
[tex]\[ x^4 + 18x^2 + 81 \][/tex]
If we consider [tex]\( x^2 \)[/tex] as a single term, this polynomial resembles the form of a quadratic equation:
[tex]\[ (x^2)^2 + 18(x^2) + 81 \][/tex]
2. Rewrite the equation in a perfect square form:
Notice that [tex]\( x^4 + 18x^2 + 81 \)[/tex] looks similar to the structure of a perfect square trinomial. A perfect square trinomial follows the form:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
By recognizing this form, we can express the original polynomial as a square of a binomial.
3. Find the suitable binomial:
Let's express the given polynomial as:
[tex]\[ x^4 + 18x^2 + 81 = (x^2 + 9)^2 \][/tex]
Here, we see:
- [tex]\( a = x^2 \)[/tex]
- [tex]\( b = 9 \)[/tex]
Using this combination:
[tex]\[ (x^2 + 9)^2 \][/tex]
4. Verify the expanded form:
To ensure correctness, let's expand [tex]\( (x^2 + 9)^2 \)[/tex]:
[tex]\[ (x^2 + 9)^2 = (x^2 + 9)(x^2 + 9) \][/tex]
Expanding this:
[tex]\[ (x^2 + 9)(x^2 + 9) = x^4 + 9x^2 + 9x^2 + 81 \][/tex]
[tex]\[ = x^4 + 18x^2 + 81 \][/tex]
Thus, we can confirm that:
[tex]\[ x^4 + 18x^2 + 81 = (x^2 + 9)^2 \][/tex]
5. Result:
Finally, the polynomial [tex]\( x^4 + 18x^2 + 81 \)[/tex] can be rewritten as a perfect square:
[tex]\[ (x^2 + 9)^2 \][/tex]
6. Simplification:
Simplifying, we state:
[tex]\[ x^2 + 9 \][/tex]
Therefore, the detailed solution is:
[tex]\[ x^4 + 18x^2 + 81 = (x^2 + 9)^2 \][/tex]
And the simplified expression is:
[tex]\[ x^2 + 9 \][/tex]
