Question

match each polynomial function to its graph.
$f(x) = -x^3 - 3x^2 - x - 3 = -(x + 3)(x^2 + 1)$
$g(x) = -x^3 + 3x^2 = -x^2(x - 3)$
$f(x) = -x^3 - 3x^2 - x - 3$
$g(x) = -x^3 + 3x^2$

Explanation:

Step1: Find zeros of $f(x)$

$f(x) = -(x+3)(x^2+1)$, so zero at $x=-3$ (only real zero).

Step2: Find zeros of $g(x)$

$g(x) = -x^2(x-3)$, so zeros at $x=0$ (double root) and $x=3$.

Step3: Match zeros to graphs

Left graph has one real zero ($x=-3$) → $f(x)$; right graph has zeros at $0$ and $3$ → $g(x)$.

Answer:

$f(x) = -x^3 - 3x^2 - x - 3$ matches the left graph; $g(x) = -x^3 + 3x^2$ matches the right graph.