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### Factoring Polynomials: Tutorial

Question:

Factor the polynomial expression [tex]$x^4+18x^2+81$[/tex] to determine the missing terms.

Drag the expressions to the correct locations on the image. Not all expressions will be used.

[tex]
\begin{array}{l}
x^4+18x^2+81=(\quad)^2+2(\quad)x^2+\left((\quad)^2\right)^2 \\
x^4+18x^2+81=(\quad)^2 \\
x^2+9 \quad x^2-9 \quad x^2 \quad 3^2 \quad 3 \quad 81 \quad x
\end{array}
[/tex]

Sagot :

GinnyAnswer
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
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