Question

factor the polynomial expression $x^4 + 18x^2 + 81$ to determine the missing terms. drag the expressions to the correct locations on the image. not all expressions will be used. $x^4 + 18x^2 + 81 = (\quad)^2 + 2(\quad)x^2 + ((\quad)^2)^2$ $x^4 + 18x^2 + 81 = (\quad)^2$ $x^2 + 9$ $x^2 - 9$ $x^2$ $3^2$ $3$ $81$ $x$ submit

Explanation:

Step1: Recognize perfect square form

Recall the perfect square formula: $(a+b)^2 = a^2 + 2ab + b^2$. For $x^4+18x^2+81$, note $x^4=(x^2)^2$ and $81=9^2=(3^2)^2$.

Step2: Match terms to the formula

Compare $x^4+18x^2+81$ to $a^2+2ab+b^2$. Here, $a=x^2$, so $2ab=2(x^2)(9)=18x^2$, and $b=9=3^2$.

Step3: Fill in the blanks

Substitute into the given structure, then write the final squared form.

Answer:

$x^4 + 18x^2 + 81 = (\boldsymbol{x^2})^2 + 2(\boldsymbol{9})x^2 + ((\boldsymbol{3})^2)^2$
$x^4 + 18x^2 + 81 = (\boldsymbol{x^2+9})^2$