# Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, 48.625) (73, 69.4375) (87, 43.9375) (100, 19)

The system, I believe, will look like: y1= a+bx+cx^2+dx^3+ex^4+fx^5+gx^6 I have not yet learned matrices, so solving this is quite hard for me. However, I am able to catch on quickly if your matrix answer is step-by-step.

Thank you so much!

• Anyway, your question is non-sense. Indeed, when $x=100$, $x^6=10^{12}$ and (using the Lagrange interpolation for example) you must work with $12$ significant digits; unfortunately, you have only $6$ significant digits ... – loup blanc Feb 10 '16 at 10:48

You can certainly find a seventh degree polynomial that goes through your points. Excel will find it for you if you plot the points, add trendline, and ask to see the equation. You will have to be quite lucky to find a sixth degree polynomial that fits. You can also ask Excel to find the best fitting sixth degree polynomial to your points. You can look in any numerical analysis text to see how to solve the simultaneous equations.

The system will look like you say, but the $x$'s have a subscript $1$ to match the $y$. There will be eight of those equations, one for each point, so you need eight constants to have a solution. You need an $hx^7$ term as well.

• ... although using matrices makes the calculation easier to explain. – Christopher Carl Heckman Feb 10 '16 at 5:11
• five points make a conic of degree 2. Add 4. Nine points for degree 6. – Narasimham Feb 10 '16 at 5:19
• @Narasimham: not if it is a polynomial in $x$. You are not allowed $xy$ terms and degree $n$ can only fit $n+1$ points – Ross Millikan Feb 10 '16 at 5:23
• One possibility guess; as OP said " I believe " .. :) – Narasimham Feb 10 '16 at 5:28