Question

The x-axis ranges from negative 3 to 4 in increments of 0.5. The y-axis ranges from negative 4 to 20 in increments of 2. The plot starts at approximately left parenthesis negative 1.6 comma negative 4 right parenthesis. It passes through the x-axis at negative 1.5 as it rises steeply to a turning point at left parenthesis negative 0.7 comma 16.5 right parenthesis. It descends to an inflection point on the x-axis at approximately 0.8 and gently rises to an inflection point at left parenthesis 2.1 comma 2 right parenthesis. The curve descends to a turning point on the x-axis at 3 and then rises steeply. The plot terminates just beyond left parenthesis 3.8 comma 20 right parenthesis. All values are approximate.Consider the graph of p(x). Given that (x−3)is a factor of p(x), is the multiplicity of (x−3)even or odd? Why?

Answer 1

Even. The graph touches the x-axis at x = 3 and turns (does not cross), which is the behavior of a root with even multiplicity (e.g. 2, 4, ...). An odd multiplicity root would make the curve cross the axis.