Cubics and quartics (1 Viewer)

[buchanan]

There are some Newton's method approximation questions on cubics and quartics in past HSC papers, and although it's not required in these questions, nevertheless it is possible to solve them exactly using the cubic and quartic formulae:

Solve for xεR:
1. x³-x²-5x-1=0 (4U, 1985)
2. x³-6x²+24=0 (3U, 1977)
3. x³-x²-x-1=0 (3U, 1981)
4. x³+x-1=0 (3U, 2004)
5. x⁴-x-13=0 (3U, 1969)

See if you can do them.

Answers: http://www4.tpgi.com.au/nanahcub/answers.gif

Formulae:
http://www4.tpgi.com.au/nanahcub/cubic.gif
http://www4.tpgi.com.au/nanahcub/quartic.gif

 

[buchanan]

Find the inverse of y=x³-3x²+4 for 0≤x≤2

Spoiler

Ans: y=2cos((cos^-1((x-2)/2)+4π)/3)+1 for 0≤x≤4

 

[Yip]

Wow buchanan, you seem to love the cubic and quartic formulae, I have seen you post about it numerous times before, so I thought you might find this question interesting, I can only do the first half of it, not the second half, perhaps you could enlighten me on it?

I think this question comes from an Oxford University Junior Mathematical Scholarship exam from the early 1900s.

 

[buchanan]

Have you tried integrating it?

 

[buchanan]

Here's an alternative way to do last year's 4 unit question on cubics. I bet you thought the answer was x³-20x+24. But that's too boring.

Of course, a better question would be that if two distinct points on E: y²=x³-5x+3 are A(x₁, y₁), B(x₂, y₂) with x₁≠x₂ show that if λ=(y₂-y₁)/(x₂-x₁) then there is a third point C on E through AB given by (λ²-x₁-x₂, λ(λ²-2x₁-x₂)+y₁).

Such a shame they don't ask better questions in HSC exams thesedays!

 

[buchanan]

Solution

Then let a=0.

 

[buchanan]

So why is it a better question?

Because this happy fact is used extensively in the arithmetic of elliptic curves and you can read more about it in

Silverman, J. H. and Tate, J., Rational Points on Elliptic Curves, Springer, 1992.

 

[buchanan]

except in semi-trivial cases, none of the methods available (including your own buchanan) are especially fast or easy (both to use and remember), and all are plagued by the possibility of numerical errors. These formulae exist, in my opinion, primarily for computer algorithms.

It's just substituting numbers into formulas. Year 7 level. Easy!. Fun! 1337!

We don't need to remember them. That's why we have the internet. Just google it.

And all you need to remember is that it's Easy! Fun! 1337!