alex asked in Science & MathematicsMathematics · 1 month ago

# how can you determine the orders of each zero in a factored polynomial equation just by looking at a graph? ?

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• 1 month ago

I will yield to a mathematician,  but I don't think you can.  The plot of (x-5)^11 looks a lot like (x-5)^13.

However, you can tell whether a zero is of odd order by seeing whether the graph crosses the x-axis, or just touches it.  A double zero, for example, will not change the sign of the graph, and a single zero will.

• 1 month ago

If the maximum order of any root is no more than about 5 you can use a few standard observations about the graph of a polynomial:

If the graph crosses the x-axis at the root, the order is odd.  If it touches and turns back, the degree is even.

If the tangent to the curve at the root is horizontal, the order is greater than 1.  If the tangent line makes a nonzero angle with the x-axis, then the order is 1.

Those will help you up to order 3.  For a multiplicity of 4 or more, the curve is noticeably flattened.   A good way to get a feel for this flattening is to compare graphs of y=x^4 and y=x^2; and the compare graphs of y=x^3 and y=x^5.  Use a site like desmos.com or wolframalpha.com if you don't have a graphing calculator handy.

If you can distinguish higher orders than 4 or 5, then you have a better eye than I do.