Discriminant. (1 Viewer)

[seanieg89]

Attached is the Q8 of an exam I wrote for a student, some of you may find it interesting...enjoy .

View attachment polynomialsexamq.pdf

 

[Carrotsticks]

Attached is the Q8 of an exam I wrote for a student, some of you may find it interesting...enjoy .

View attachment 24019

Isn't the discriminant of the cubic that identity that we always see in the Extension 2 paper: 27q^2+4p^3?

I'm going out now, but I'll give it a try when I get back.

 

[hayabusaboston]

wow nice mx1 and mx2 results for hsc, you'e the first guy ive seen who got above 98 for mx2, besides the 200/200 dude on the kids science tv show with karl kruszelnicki.

 

[seanieg89]

Yep. This is an algebraic derivation of that result that generalises to higher degree polynomials (although it turns out that the discriminant is not as useful for them).

 

[RealiseNothing]

wow nice mx1 and mx2 results for hsc, you'e the first guy ive seen who got above 98 for mx2, besides the 200/200 dude on the kids science tv show with karl kruszelnicki.

largarithmic from this forum got 99 in mx2 iirc.

 

[Carrotsticks]

Regarding the first part, is alpha_l some arbitrary complex root (or real, we don't know) for some k < l < n ?

Also regarding the product notation where you wrote k < l, again, is this for some arbitrary k < n, or is this related to the summation notation starting from k=0?

 

[seanieg89]

Regarding the first part, is alpha_l some arbitrary complex root (or real, we don't know) for some k < l < n ?

Also regarding the product notation where you wrote k < l, again, is this for some arbitrary k < n, or is this related to the summation notation starting from k=0?

Every is a complex root of P. By the FTOA, there are exactly n such roots, counting multiplicity.

The convention here is to take the sum/product over all pairs of positive integers (k,l) with .

(This is a fairly common notation for sums and products.)

 

[Carrotsticks]

The convention here is to take the sum/product over all pairs of positive integers (k,l) with .

That's what I wanted to confirm. Thanks.

 

[largarithmic]

Regarding the first part, is alpha_l some arbitrary complex root (or real, we don't know) for some k < l < n ?

Also regarding the product notation where you wrote k < l, again, is this for some arbitrary k < n, or is this related to the summation notation starting from k=0?

this usually would mean the product taken over all k < l right?

so like, for a quadratic with roots a1 and a2, its (a1-a2)^2; for a cubic roots a1,a2,a3 its [(a1-a2)(a1-a3)(a2-a3)]^2, for a quartic its [(a1-a2)(a1-a3)(a1-a4)(a2-a3)(a2-a4)(a3-a4)]^2, etc so that there are n choose 2 multiplicands for a degree n polynomial.

EDIT: post got beaten

Oh and really neat question ^^ Especially the first part

 

[Carrotsticks]

Hold on, did you mean [k,l) instead of (k,l) ?

I've finished all the other parts, but I'm just getting all confused about the notation for the first lol.

I've never done a question utilising the product notation with an inequality like that at the bottom, usually just ones where it's the same as summation notation, except multiplying.

EDIT: Finished first part. Very nice question. However, the notation may be a bit alien to Extension 2 students.

 

[largarithmic]

Hold on, did you mean [k,l) instead of (k,l) ?

I've finished all the other parts, but I'm just getting all confused about the notation for the first lol.

I've never done a question utilising the product notation with an inequality like that at the bottom, usually just ones where it's the same as summation notation, except multiplying.

whaaa how does [k,l) change anything? its not interval notation, its just ordered pair notation Im pretty sure...

you can do that notation for summation too (it's shorthand for writing 1 <= k < l <= n), e.g.



and if those a_i were roots of a monic degree n polynomial, that would be equal to negative the x^n-3 coefficient

 

[seanieg89]

Exactly what larg said, we are just talking about ordered pairs of integers. An alternate way of writing the same product is:



Thanks for the positive feedback on the question, I felt it might help students appreciate the discriminant as more than just a random expression that happens to pop up under a square root sign in the quadratic formula.


Note:
5) is somewhat tedious but I feel there are enough algebraic shortcuts available to the clever student that it is worth testing.

 

[lolcakes52]

Okay, as stupid as it sounds. I am finding the Pi notation absolutely mind destroying, can you give a simple example and I will probably be able to figure it out.

 

[deswa1]

Sigma notation refers to addition, the pi refers to multiplication. For example (I don't know how to make this neat) but if you had n=1 on the bottom of the pit, n=8 on the top and n^2 on the side, it means (1^2)x(2^2)x(3^2)x...x(8^2). Does that make sense?

 

[Carrotsticks]

Sigma notation refers to addition, the pi refers to multiplication. For example (I don't know how to make this neat) but if you had n=1 on the bottom of the pit, n=8 on the top and n^2 on the side, it means (1^2)x(2^2)x(3^2)x...x(8^2). Does that make sense?

That's for normal Pi notation. This one is different.

Okay, as stupid as it sounds. I am finding the Pi notation absolutely mind destroying, can you give a simple example and I will probably be able to figure it out.

Remember in Polynomials when we did Sum of roots, product of roots etc?

Remember how you could do the sum of roots in pairs? c/a ? it was AB + BC + AC ? We used up all the different possible combinations of pairs?

This is the exact same thing, but instead of adding them, we're multiplying them.

As a result, we should have C(n,2) expressions.

 

[seanieg89]

Okay, as stupid as it sounds. I am finding the Pi notation absolutely mind destroying, can you give a simple example and I will probably be able to figure it out.

For n=3,

 

[lolcakes52]

For n=3,

Thank you kind sir!