Question

match each polynomial function to its graph.
$f(x) = x^3 + 22x^2 + 160x + 384 = (x + 8)^2(x + 6)$
$g(x) = x^3 + 21x^2 + 147x + 343 = (x + 7)^3$
$f(x) = x^3 + 22x^2 + 160x + 384$ $g(x) = x^3 + 21x^2 + 147x + 343$
two graphs of polynomial functions are shown, each with a coordinate grid from -10 to 10 on both axes. the first graph has a curve that touches the x-axis at x = -8 and crosses at x = -6. the second graph has a curve that crosses the x-axis at x = -8 and x = -6 with a local maximum between them. there are empty boxes below each graph for matching the functions.
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work it out

Explanation:

Step1: Identify roots of \(f(x)\)

\(f(x)=(x+8)^2(x+6)\), roots: \(x=-8\) (double), \(x=-6\)

Step2: Identify roots of \(g(x)\)

\(g(x)=(x+7)^3\), root: \(x=-7\) (triple)

Step3: Match roots to graphs

Left graph has a root at \(x=-6\) and a touch at \(x=-8\) (double root); Right graph has a flat root at \(x=-7\) (triple root)

Step4: Assign functions to graphs

Left graph → \(f(x)\); Right graph → \(g(x)\)

Answer:

Left graph: \(f(x) = x^3 + 22x^2 + 160x + 384\); Right graph: \(g(x) = x^3 + 21x^2 + 147x + 343\)