Alan G. answered • 07/17/16
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Nicole,
When a polynomial has a zero or x-intercept, it occurs because there is a factor for that zero. For example, if 6 is a zero of a polynomial p(x), it will have at least one factor (x − 6). The multiplicity of that zero is the number of times the factor appears in the polynomial when it is factored completely.
Another example. Let p(x) = (x − 6)5(x + 4)4. The zeros are 6 and −4. the zero 6 has multiplicity 5 and the zero −4 has multiplicity 4.
The parity of the multiplicity (whether it is even or odd) determines how the graph of p(x) meets the x-axis at the zero.
In this last example, the graph will cross the x-axis at x = 6, and the graph will look a lot like y = x5 as it goes through the origin. The graph will also just touch (and not cross) the x-axis at x = −4, and look a lot like y = x4 as it passes through the origin. If you have a graphing calculator, you can graph the function and see this quickly.
The rule is this:
When the zero has odd multiplicity, the graph will cross the x-axis at the zero. When the zero has even multiplicity, the graph will touch the x-axis at the zero.
I have given you a single example which illustrates both behaviors, so I hope this will be enough. You can ask more questions if it is not.
Alan G.
Kenneth,
The graph of y = x^5 flattens as it passes through the origin. It does cross the x-axis there and the mutliplicity of the zero is odd. Similarly, the graph of y = x^4 flattens as it goes through the origin but does not cross the x-axis there, but only touches it. The word for this behavior is "tangent," which is a crucial concept in calculus and geometry.
This is what I meant when the graph " looks a lot like" these at their zeros.
Let me know whether you require more explanation on this. The best way to see what I am trying to say is to graph the functions p(x) and these two power functions and compare the results.
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07/18/16
Kenneth S.
07/17/16