Sure, let's go through the steps to factor the polynomial [tex]\(x^4 + 18x^2 + 81\)[/tex].
1. Identify Recognizable Patterns:
- The given polynomial is [tex]\(x^4 + 18x^2 + 81\)[/tex].
- Notice that this expression resembles a perfect square trinomial. Perfect square trinomials take the form [tex]\(a^2 + 2ab + b^2 = (a + b)^2\)[/tex].
2. Identifying [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- Compare [tex]\(x^4 + 18x^2 + 81\)[/tex] with [tex]\(a^2 + 2ab + b^2\)[/tex].
- Here, let's identify the terms:
- [tex]\(a^2 = x^4\)[/tex], which implies [tex]\(a = x^2\)[/tex].
- [tex]\(b^2 = 81\)[/tex], which implies [tex]\(b = 9\)[/tex] (since [tex]\(9^2 = 81\)[/tex]).
- Therefore, we can recognize [tex]\(x^4 + 18 x^2 + 81\)[/tex] as a perfect square trinomial: [tex]\((x^2 + 9)^2\)[/tex].
3. Rewrite the Polynomial:
- Rewrite the polynomial in its factored form:
[tex]\[
x^4 + 18 x^2 + 81 = (x^2 + 9)^2.
\][/tex]
4. Decompose the Polynomial:
- To match the form given in the tutorial question:
[tex]\[
x^4 + 18 x^2 + 81 = (x^2)^2 + 2(x^2)(9) + (9)^2.
\][/tex]
- Notice that the expressions inside correspond to:
- The first blank ([tex]\(\quad\)[/tex]) should be [tex]\(x^2\)[/tex].
- The second blank ([tex]\(\quad\)[/tex]) should also be [tex]\(x^2\)[/tex].
- The third blank ([tex]\(\quad\)[/tex]) should be 9.
- Therefore,
[tex]\(
x^4 + 18 x^2 + 81 = (x^2)^2 + 2(x^2)(9) + (9)^2.
\)[/tex]
5. Final Factored Form:
- Thus, the completed form:
[tex]\(
x^4 + 18 x^2 + 81 = (x^2 + 9)^2.
\)[/tex]
So, filling in the blanks correctly:
- For the equation [tex]\(x^4 + 18 x^2 + 81 = (\quad)^2 + 2(\quad) x^2 + \left((\quad)^2\right)^2\)[/tex]:
- First blank: [tex]\(x^2\)[/tex].
- Second blank: [tex]\(x^2\)[/tex].
- Third blank: [tex]\(9\)[/tex].
- For the equation [tex]\(x^4 + 18 x^2 + 81 = (\quad)^2\)[/tex]:
- Blank: [tex]\(x^2 + 9\)[/tex].
So, using the given expressions, you would correctly fill in the blanks as follows:
- [tex]\(x^4+18 x^2+81=(x^2)^2+2(x^2) x^2+\left(9^2 \right)^2\)[/tex]
- [tex]\(x^4+18 x^2+81=(x^2 + 9)^2\)[/tex]
To drag the expressions:
[tex]\(\boxed{x^2}\)[/tex], \boxed{9}, \boxed{x^2 + 9}\quad
