Question

match each polynomial function to its graph.
$f(x) = -x^4 - 5x^2 - 4 = -(x^2 + 4)(x^2 + 1)$
$g(x) = -x^4 + 4x^3 - 6x^2 + 8x - 8 = -(x - 2)^2(x^2 + 2)$
two polynomial functions and two graphs are shown, with empty boxes below each graph to match the functions.

Explanation:

Step1: Analyze \(f(x)\)

\(f(x) = -(x^2 + 4)(x^2 + 1)\) has no real roots (since \(x^2 + 4 > 0\) and \(x^2 + 1 > 0\) for all real \(x\)), so its graph never crosses the x-axis. The left graph matches this.

Step2: Analyze \(g(x)\)

\(g(x) = -(x - 2)^2(x^2 + 2)\) has a real root at \(x = 2\) (double root), so its graph touches the x-axis at \(x = 2\). The right graph matches this.

Answer:

Left graph: \(f(x) = -x^4 - 5x^2 - 4\); Right graph: \(g(x) = -x^4 + 4x^3 - 6x^2 + 8x - 8\)