polynomials... (1 Viewer)

[queenie]

found a couple of 'exploratory excersises' in a book somewhere.. (i wouldnt have a clue whether this is 3U or 2U, so mods, feel free to move this)

ok:

1) a) Show that the sub t=x + 3 transforms the cubic polynomial
y= x^3 - 9x^2 + 31x -44 into the form y=t^3 +at + b
b) Using a) or otherwise, show that the equation x^3-9x^2+31x-44 has exactly one real root.

2) Find a substitution which transforms the cubic y=x^3 - x^2 +5x-2 to the form y = t^3 + ct + d.

3) a) Verify that the line y=6x+21 is tangential to the cubic y=x^3 - 6x +5 at the point P(-2, 9).
b) The tangent at P in a) meets the cubic again at Q. Find the co-ords of Q.

i did most of them, but i want to check my answers with some1..

 

[nike33]

3a) y = 6x+21, y = x^3 - 6x + 5..solve simul

6x+21 = x^3 - 6x + 5
f(x) = x^3-12x-16 (representing points of intersection)
f(-2) = 0
f'(x) = 3x^2 - 12
f'(-2) = 0

as f(x) has a double root at x = -2 and f(x) rep point of interection..hence two lines at x=-2 is a tangent

 

[nike33]

2) x = (t + 1/3)

as when expanding the cubic the 3C2 will cancel with the 1/3 getting x^2 - x^2

 

[kpq_sniper017]

Originally posted by queenie
found a couple of 'exploratory excersises' in a book somewhere.. (i wouldnt have a clue whether this is 3U or 2U, so mods, feel free to move this)

ok:

1) a) Show that the sub t=x + 3 transforms the cubic polynomial
y= x^3 - 9x^2 + 31x -44 into the form y=t^3 +at + b
b) Using a) or otherwise, show that the equation x^3-9x^2+31x-44 has exactly one real root.

2) Find a substitution which transforms the cubic y=x^3 - x^2 +5x-2 to the form y = t^3 + ct + d.

3) a) Verify that the line y=6x+21 is tangential to the cubic y=x^3 - 6x +5 at the point P(-2, 9).
b) The tangent at P in a) meets the cubic again at Q. Find the co-ords of Q.

i did most of them, but i want to check my answers with some1..

interesting.....
i wouldn't mind having that "exploratory book".....might help me for the MANSW Mathsearch Comp.
they're the first three exercises of the Mathsearch Comp. booklet....what book was that again
any chance of lending it to me

 

[Estel]

queenie that's a very unfair thing to do...
also known as cheating...

 

[Estel]

except u did a typo on the first q rofl

 

[velox]

(i wouldnt have a clue whether this is 3U or 2U, so mods, feel free to move this) yes because its in the booklet. tisk tisk

 

[Estel]

yes, tisk tisk tisk.










You have the answers to the other questions pplz?

 

[queenie]

Originally posted by Estel


You have the answers to the other questions pplz?

im sorry, i thought it was cheating...

im actually not takin part of the mathsearch this year, one of my friends asked me the questions, and i did them but wanted to clarify...

 

[velox]

lol, that is v. low Estel.

 

[Estel]

It was actually meant to be an attempt at humour, considering I had just said tisk for someone cheating.

Pity it got so badly misinterpreted.

In any case Section A isn't that hard.

 

[kpq_sniper017]

Originally posted by Estel
It was actually meant to be an attempt at humour, considering I had just said tisk for someone cheating.

Pity it got so badly misinterpreted.

In any case Section A isn't that hard.

when did u get urs??
got mine yesterday, but i've got a lot of assessments, so i don't have much time to do it.
i think i know how to do most of part a.

did u read the "preamble" and the introduction to cubolinear stuff? lol, do we really need to know aboit structures and systems and blah.....??

 

[kpq_sniper017]

Originally posted by queenie
im sorry, i thought it was cheating...

im actually not takin part of the mathsearch this year, one of my friends asked me the questions, and i did them but wanted to clarify...

in that case, i guess ur friend's cheating....tsk tsk.
and even so, u said "found a couple of 'exploratory excersises' in a book somewhere..".....y not just say: "my friend needs some help with the mathsearch comp".....blah.

ah well....no more help for queenie or her friend no more

 

[kpq_sniper017]

anyhow, if ur friend can't do exercise 1 without assistance......i don't know how he/she is gonna go with the rest of it.

 

[Estel]

I got the stuff yesterday. I read through the preamble and it was all new to me... having attempted (and failed) to do the enrichment in past years without reading any of the given material, I decided to do things differently this time.

Structures... pfft glossed over that. The polynomial stuff about roots and tangents is important though.

It took a few hours last night to finish section A and rewrite it neatly. If anyone isn't willing to spend that time, instead of asking for help, they shouldn't do the competition.

pcx I'm sure you can do it faster than I did (you're in year 12?), so it won't really interrupt your assessments too much.

 

[kpq_sniper017]

Originally posted by Estel
I got the stuff yesterday. I read through the preamble and it was all new to me... having attempted (and failed) to do the enrichment in past years without reading any of the given material, I decided to do things differently this time.

Structures... pfft glossed over that. The polynomial stuff about roots and tangents is important though.

It took a few hours last night to finish section A and rewrite it neatly. If anyone isn't willing to spend that time, instead of asking for help, they shouldn't do the competition.

pcx I'm sure you can do it faster than I did (you're in year 12?), so it won't really interrupt your assessments too much.

i'm in yr11... the mansw comp. is only for yr11'ers i think. the two weeks i have actually fall right on the two weeks when i have the most assessments.
so, if i don't get enough done....well i'm out of the comp
how many marks out of the 135 do u think u need to get to win the comp.?

 

[Estel]

To be honest, 135 or quite close.
If it continues in vein of section A I can see a lot of people getting 135.

If not, some cheater who gets stuff from mathematic teacher parents or a JR band of friends (I would include other schools, I'm not singling you guys out, it's just you're the only ones with enough smart math kids in one place) will do well.

 

[kpq_sniper017]

Originally posted by Estel
To be honest, 135 or quite close.
If it continues in vein of section A I can see a lot of people getting 135.

If not, some cheater who gets stuff from mathematic teacher parents or a JR band of friends (I would include other schools, I'm not singling you guys out, it's just you're the only ones with enough smart math kids in one place) will do well.

135 - seems pretty tough.
i wonder if anybody has honestly won the comp ie. without any outside assistance at all.

 

[velox]

only the top 4ish ppl in our yr get to do it. and its hard to be one of them considering they have all finished 4u already i wanted to do it

 

[Estel]

Heh in our school anyone in Yr 11 or the top class in Yr 9/10 gets to do it. I don't think anyone in Yr 9/10 actually hands it in tho'.