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Question

A plotted curve with 3 inflection points is labeled m left parenthesis x right parenthesis. The plot starts in quadrant 2, descends to the x-axis, rises to a turning point in quadrant 1, descends to a turning point in quadrant 4 and rises through quadrant 1.

Consider the graph of m(x). Given that (x−4)
is a factor of m(x), is the multiplicity of (x−4)
even or odd?

Option #1: even

Option #2: odd

(1 point)
The multiplicity of (x−4)
is described by Option #
.

Answer 1

To determine the multiplicity of the factor \((x - 4)\) in the polynomial \(m(x)\), we need to consider the behavior of the graph at \(x = 4\):

1. If \((x - 4)\) has an even multiplicity, the graph will touch the x-axis at \(x = 4\) and turn back without crossing it.
2. If \((x - 4)\) has an odd multiplicity, the graph will cross the x-axis at \(x = 4\).

Given the described behavior of the graph \(m(x)\) where it starts in quadrant 2 (above the x-axis), descends to the x-axis and then rises again (implying the graph passes through the x-axis), the factor \((x - 4)\) must have odd multiplicity, because the graph crosses the x-axis at this point.

Thus, the multiplicity of \((x - 4)\) is described by Option #2: odd.