Fresh C.

asked • 09/22/17

suppose there is n = 40 points in the plane, each with a distinct x coordinate.

 What is the smallest degree d such that there is always polynomial of degree d whose graph passes through all n points?

1 Expert Answer

By:

Arturo O. answered • 09/22/17

Experienced Physics Teacher for Physics Tutoring

Fresh C.

would that mean d= 40?
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09/22/17

Fresh C.

the options given for an answer were: 40, 41, 42, 39. this is what confused me initially
 
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09/22/17

Arturo O.

For this problem, as currently worded, the lowest degree is d = 0.  I suspect the problem is not worded correctly.  Perhaps the original problem said distinct y coordinates?  Please double check.
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09/22/17

Arturo O.

There just is not enough information given in the problem statement.  You could have a straight line passing through the points (degree 0 or 1), a parabola (degree 2), a cubic (degree 3), etc.  The answer depends on how the points are arranged.
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09/22/17

Fresh C.

I copied the original problem word for word. your explanation makes a lot of sense! but "0" isn't one of the options for an answer. i'm not sure how the numbers 39,40,41,42 make sense as answers in this context?
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09/22/17

Arturo O.

I would consider the following extreme case:
 
The points are arranged so that from left to right, the first one is below the x-axis, the second is above the x-axis, the third is below the x-axis, etc.  If you plot 3 points like this, you need to cross the x-axis twice, meaning the degree is at least 2.  If you plot 4 points like this, you need to cross the x-axis 3 times, meaning the degree is at least 3.  Plot 5 points, and you need to cross the x-axis 4 times.  Do you see the pattern?  It looks like if you plot 40 points, you need to cross the x-axis 40-1=39 times.  The degree is the number of times you cross the x-axis, in this extreme case.  Then the answer to your question is 39.  Since this is an extreme case, then degree 39 guarantees all other possible arrangements of 40 points.  Is this an on-line exercise?  If so, try entering 39 for the answer and see if it accepts it.
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09/22/17

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