Question

select the correct answer.
which polynomial is equivalent to $f(f(x))?$
$f(x) = -x^2 + 3x$
$\bigcirc$ $x^4 - 6x^3 - 12x^2 + 9x$
$\bigcirc$ $-x^4 - 6x^3 - 9x^2 + 3x$
$\bigcirc$ $-x^4 + 6x^3 - 12x^2 + 9x$
$\bigcirc$ $x^4 - 9x^2 + 3x$

Explanation:

Step1: Substitute $f(x)$ into itself

$f(f(x)) = -(-x^2 + 3x)^2 + 3(-x^2 + 3x)$

Step2: Expand $(-x^2 + 3x)^2$

$(-x^2 + 3x)^2 = (-x^2)^2 + 2(-x^2)(3x) + (3x)^2 = x^4 - 6x^3 + 9x^2$

Step3: Substitute back and simplify

$f(f(x)) = -(x^4 - 6x^3 + 9x^2) - 3x^2 + 9x = -x^4 + 6x^3 - 9x^2 - 3x^2 + 9x$

Step4: Combine like terms

$f(f(x)) = -x^4 + 6x^3 - 12x^2 + 9x$

Answer:

$-x^4 + 6x^3 - 12x^2 + 9x$ (the third option)