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## Monday, January 23, 2017

### Choosing from 4 wrong answers?

The question below is from the June 2016 Algebra II regents exam in New York. Some questions have been raised about it claiming it might confuse those who are unsure as to how to "count" multiple roots. (For clarity, a multiple root is a crossing root if the root occurs an odd number of times, and a tangent root if it occurs and even number of times.)

Actually, that issue is irrelevant in this question. Based on the accepted standard regarding the meaning of "arrows" at the end of the graph, choice 3 is the only choice satisfying the second and third bulleted items. The first bulleted item is irrelevant in this question.

Or is it? It is possible that choice 3 has at least one root of odd multiplicity of 3 or more, but without a scale on the graph it is impossible to tell.

Or is it? Something did not seem right about choice 3. (Continued below)

I had to do some investigating.

In GeoGebra I created a file including the graphic from choice 3 along with a cubic polynomial graph sharing the x-intercepts with those in choice 3.  Since there were no scales on the axes in any of the choices, the only information I could rely on was the relative positions of key points.

Here is the GeoGebra sketch (get it here if you wish):

The dotted blue line marks midway between the leftmost x-intercepts.

No matter what I do to change what is changeable (experiment with changing leading coefficient or absolute position of leftmost root) the polynomials maximum and choice 3's maximum lie on opposite sides of that line.

I will not claim that I have the definitive answer on this issue, but it appears to me that the more you know about cubic polynomial graphs the less likely you are to accept choice 3 as an answer to this question.  The problem could have been avoided if actual polynomial graphs had been given.

I believe I understand what the question writers intended with this question, but I must reject the question itself.