Lesson 2 of 3
Predict the ends and use multiplicity to explain crossing or touching.
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Worked example
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For , the right end rises and left end falls. Its zero at 0 has multiplicity 3, so it crosses the axis.
Big ideas
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For very large positive or negative x, a polynomial behaves like its leading term . Use this to describe end behavior; it does not locate every turning point.
For even n, the ends point the same way: both up if , both down if . For odd n, gives left down and right up; reverses these.
A real zero with odd MultiplicityHow many times the same zero appears as a factor.Exactly: A zero has multiplicity when divides the polynomial but does not.Mix-up alert: Even multiplicity touches the axis; odd multiplicity crosses it.See this in Key words crosses the x-axis. A zero with even multiplicity touches the axis and turns back. This describes sign changes near that zero.
A degree n polynomial has at most turning points. This is an upper bound, not an exact count. Zeros and end behavior alone may not fix the whole graph.
Worked example
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For , degree is 3 and leading coefficient is negative. The left end rises and right end falls. At it touches; at 3 it crosses. At most two turning points are possible; that bound is not an exact count.
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Describe the ends and x-axis behavior of .
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Worked example
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For , the leading term is . A large positive or negative input gives a large positive sixth power, multiplied by -4. Thus both ends fall: p(x) tends to -infinity as x tends to either infinity or -infinity.
Worked example
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The cubic has degree 3, so at most 2 turning points. It actually has none: as x increases, keeps increasing. Its zero at 0 has multiplicity 3, and the graph crosses the axis there without turning.
Mix-up alert
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Wrong idea: every zero is a turning point. Repair: an odd-multiplicity zero crosses the axis; an even-multiplicity zero touches and turns back. Some turning points occur away from the x-axis.
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Even powers are positive on both far sides.
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How do the ends of a positive-leading even-degree polynomial behave?
Both rise.
How do the ends of a negative-leading odd-degree polynomial behave?
Left rises, right falls.
What happens at a multiplicity-2 real zero?
The graph touches the x-axis and turns back.
What happens at a multiplicity-3 real zero?
The graph crosses the x-axis.
How many turning points can degree 6 have at most?
5.
Does a cubic have to have two turns?
No. Two is only the maximum.
Which term controls end behavior in ?
.
Does an even-multiplicity zero change the local sign of the polynomial?
No.
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Self-test
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