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Polynomial Interpolation

Age 16 to 18 Challenge Level:

Why do this problem?

This problem introduces students to interpolation and the concept of 'building' algebraic solutions to problems. The result is very interesting mathematically. It is based upon the idea of a 'generic example': a particular example which encapsulates in a clear way all of the properties of a more general case. The ideas in this problem pave the way for patterns of thinking which are to be found in university mathematics courses, and the concept of interpolation is valuable in both mathematics and science. This problem also raises the idea from proof that constructing an example with the correct properties proves that an example exists, whereas inability to construct an example with the correct properties does not necessarily prove that such an example doesn't exist.


Possible approach

This problem requires that students understand the concept of fitting a curve through a set of points to mean, roughly, 'write down a function $y=f(x)$ which passes exactly through each of the specified points', and will need to understand that we are not interested in the function anywhere except at these isolated points.
Once this is understood you can simply put the first part of the question on the board and pose the question 'does this solve Steve's problem?'. The algebraic expression might initially appear intimidating or mysterious, but gradually students will realise that most of the expression evaluates to zero at each of the points and then will realise why the numbers have been written as they have been.
It is important that students be given enough time to grapple with the first algebraic expression as by doing this they will appreciate the structure which leads to it 'obviously' solving the problem. You can do this in small groups or as a whole class-based discussion (you can use the Key Questions to help with this discussion). Be sure to do this without pencil and paper in the first instance, because some students might be tempted to expand out all of the brackets and this destroys the structure of the construction.
Once this expression is seen to solve Steve's problem, a brief discussion can be had concerning why Steve wrote the expression as he did. For example, why is the whole expression not expanded? Why not simplify some of the brackets such as $(1-2)$?
Now students should be ready to tackle the rest of the problem on their own. Clear critical thinking is required, and if students believe that certain results are 'obvious' be sure to get them to explain clearly what is so obvious: students will need to pay particular attention to zeros occurring in their construction.
For the final part it is worth asking students to consider: does a quadratic fit through the five points? How do you know (if you claim that you do) either way?
Finally, some students might really want to simplify the expressions to find the coefficients of the polynomials or to plot the graphs to see the shapes of the answers. This is fine if they want to!

Key questions

What does Steve's expression evaluate to at each of the three points?

Why has Steve not expanded the brackets or simplified the numbers?

Do you think that Steve's construction could be generalised?

Under which circumstances would Steve's construction break down?

How many degrees of freedom does a quadratic polynomial have?


Possible extension

There are some extension possibilities in the question. Other extension possibilities are to use a spreadsheet or computer to produce the fitting polynomials for $4$ or $5$ variable points. This is mathematically very interesting and will lead to the understanding that the fitting polynomials, whilst exactly hitting all of the points, are often very unstable and with wildly varying shape - this is a good open investigation which might arise from this problem.


Possible support

The main idea in this question concerns constructing a fitting polynomial by arranging brackets and coefficients in a sensible, organised manner. This key idea can be practised by looking at fitting quadratics through 3 points: give a few triples of points and have students construct the quadratics which go through these.