Question

match each polynomial function to its graph.
$f(x) = x^4 + 12x^3 + 54x^2 + 108x + 81 = (x + 3)^4$
$g(x) = x^4 + 5x^2 + 4 = (x^2 + 1)(x^2 + 4)$
$f(x) = x^4 + 12x^3 + 54x^2 + 108x + 81$
$g(x) = x^4 + 5x^2 + 4$

Explanation:

Step1: Analyze $f(x)=(x+3)^4$

It has a repeated root at $x=-3$ (multiplicity 4), so the graph touches the x-axis at $x=-3$ and turns around. The left graph has a turning point at $x=-3$, consistent with this.

Step2: Analyze $g(x)=(x^2+1)(x^2+4)$

All factors are quadratic with no real roots (since $x^2+1=0$ and $x^2+4=0$ have imaginary solutions), so the graph never crosses the x-axis. The right graph does not intersect the x-axis, consistent with this.

Answer:

$f(x)=(x+3)^4$ matches the left graph; $g(x)=(x^2+1)(x^2+4)$ matches the right graph.