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effort
Senior Member

USA
38 Posts

 Posted - 12/01/2011 :  11:18:47 Solve each equation. State the number and type of roots.1. (x^4) + 625 =0 2. (x^5) +4x^ = 01. I solved for x and did get 4 answers: 5i, -5i, 5i, -5i four complex root.2. I solved for x and did get 5 answers: 0,0,0, 2i, -2iHow do you explain that o is an answer 3 times. And how do you explain that the graph goes through the zero 3 times.

royhaas
Moderator

USA
3059 Posts

 Posted - 12/01/2011 :  14:00:01 Try graphing 2.

the_hill1962

USA
1469 Posts

 Posted - 12/06/2011 :  11:55:23 This is something that I have also wondered about. I know that a "double root" touches the x axis but doesn't cross. a root that is repeated an odd number of times does cross.effort: You are asking for an explanation. I don't know if I am correct in the following, but here is how I think about it---The graph of the curve goes "in" and and instead of just continuing through it, it goes "out" on the same side it came in. Therefore I just think, "that is why there are two roots there".You have a good question asking for an explanation of a triple root!I have the same question.Is there anyone that can offer an "explanation"?

royhaas
Moderator

USA
3059 Posts

 Posted - 12/06/2011 :  13:28:31 Multiple real roots only affect the shape of the graph. Compare the graphs of y=x^3-x with y=x^4-x^2 over the interval [-2,2]. The overall shape of a graph also depends on first and second derivatives. Note that the distinct roots in both cases is {-1,0,1}. The degree of the polynomial also contributes heavily to the overall shape, as evidenced by the graphs of x^2-1 with x^4-1. Each has two real roots but teir shapes are different in [-1,1].

effort
Senior Member

USA
38 Posts

 Posted - 12/06/2011 :  13:51:39 Hill, yes, that is my question. Thank you for understanding my question.

the_hill1962

USA
1469 Posts

royhaas
Moderator

USA
3059 Posts

 Posted - 12/07/2011 :  06:34:43 The "flattening" occurs as a result of the behavior of the polynomial in |x|<1. This is the case even if all roots are conjugate imaginary roots. The behavior becomes more pronounced with the degree of the polynomial. For example, look at (x^2+1)^2 vs (x^2+1)^4, both of which have repeated complex roots but no real zeroes.
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