# Polynomial interpolation

Steve wishes to draw a quadratic polynomial $y(x) = ax^2+bx+c$ through the three points $(x, y)=(1,4), (2, 5), (3, 7)$. He writes down this expression:

$$y(x) = 4\frac{(x-2)(x-3)}{(1-2)(1-3)}+5\frac{(x-1)(x-3)}{(2-1)(2-3)}+7\frac{(x-1)(x-2)}{(3-1)(3-2)}$$

Does this solve the problem that Steve was trying to address?

Using Steve's example as a guide, can you construct a quadratic polynomial which passes through the three points $(1,2), (2, 4), (4, -1)$? Is this the only such quadratic polynomial?

Can you construct a cubic polynomial which passes through the four points $(1,2), (2, 4), (3, 7),(4, -1)$. Is this the only such cubic polynomial?

By this stage, the particular examples you have constructed should give you ideas about how to construct a 'general' case. Use your insights to answer these questions:

Can you write down an expression for a line passing through the two points $(2, 7)$ and $(8,-6)$ using this method?

Can you always fit a quadratic polynomial through three points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$?

Can you always fit a quartic polynomial through five points $(x_i, y_i)$ ($i=1\dots 5$) where exactly two of $x_i$ are zero?

How many different polynomials can you construct which would pass through the points $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)$?

Explore sets of points through which it is not possible to fit a polynomial.

**Extension**: For various numbers of points and degrees of polynomial you might wish to consider when the fitting is unique, when it is possible with multiple polynomials and when it is impossible.

With an interpolation, we are only interested in the properties of the function at the specified interpolation points, so there is no need to worry about the form of the functions elsewhere or to sketch them (although feel free to explore this if you are interested!).

Don't be tempted to expand out Steve's expression, as this will hide the properties of the construction which allow you to see generalisations.

To get started, try substituting in a point and see what cancels or vanishes!

This problem didn't receive a solution when it was published in March 2012. Perhaps you can submit a solution?

### 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

*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.

*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.

### 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.