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The given fact will relate all of these ideas to the multiplicity of the zero.
Fact
If x = r is a zero of the polynomial P (x) along with multiplicity k then,
1. If the value of k is odd then the x-intercept corresponding to x = r will cross the x-axis
2. If value to k is even then the x-intercept corresponding to x = r will just touch the x-axis and not cross it actually.
Furthermore, if k = 1 then the graph will flatten out at the point x = r .
At last, notice that as we consider x get large in both the +ve or -ve sense (that means at either end of the graph) then the graph will either increase with no bound or decrease without bound. It will always occur with every polynomial and we can employ the following test to find out just what will happen at the endpoints of the graph.
We now can also combine the two shifts we only got done looking at into single problem. If we know the graph of f ( x ) the graph of g ( x ) = f ( x + c ) + k will be the graph of
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Now let's move into the next technique for solving systems of equations. As we illustrated in the example the method of substitution will frequently force us to deal with fraction
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