HSC Tips - Polynomials (1 Viewer)

[McLake]

OK, trials are nearing so here is installment 4

Tips for Polynomials:
- Know how to transform ploynomials when the roots are changed to the following (A is alpha, B is beta etc):
-- A + 1, B + 1, C + 1
-- A^2, B^2 ...
-- A^3, B^3 ...
-- 1/A, 1/B ...
[There is usually up to 4 roots, and there is the possibility of mixing two or more "transformations" (ie: (A + 1)^2)]

- Be prepared to use with complex nos.

- Know how to do long divsion with polys, complex polys, unknonw coeffecient polys.

- Know the answer to this question:
p(x) = x^4 + x^3 +4x^2 + 2 and one of the roots is 3 - i (this may or may not be true, it's not relevent here)

EXPLAIN why 3 + i is a root

ANSWER: "If a polynomial with REAL coefficents has a complex root, the its conjugate is also a root." (This was in last years HSC)

- Know the remainder and factor theorms.

 

[tonberry_kun]

thnx mclake
post sumore plz

 

[McLake]

Originally posted by tonberry_kun
thnx mclake
post sumore plz

More? I don't know any more for Polynomials. Intergration will come soon ...

 

[Affinity]

General
--property of rational roots of Polynomials over Z.
--Polynomials with multiple roots.
--find sum of the roots to the power of n.
--Using the fundamental theorem of algebra to prove things.
--Solving quartics with symmetric coefficients.

I am making a wild guess that there will be a harder poly q this year. -> just a hunch.

 

[ezzy85]

complex roots and roots of unity.
setting out those trig/de moivres theorem qs properly.

 

[underthesun]

I hope this HSC tips finishes before my Maths trial starts

keep up the good work

 

[McLake]

It should be done by next week ...

 

[tonberry_kun]

excellent

 

[freaking_out]

Originally posted by underthesun
I hope this HSC tips finishes before my Maths trial starts

i also hope that the main page is updated with the new resources before my trials which is in 2 weeks time!!!

but then again who am i to complain- coz i don't pay for all this anyway.

 

[underthesun]

relax, may the force be with you

 

[Affinity]

the best preparation for exams is to explain things to others. If you can do it, it means you truely understand the concepts and usually from a variety of aspects

 

[spice girl]

here is a bit more tips. may be a bit repititious, but anyway:

* if there is any mention of multiple roots, be prepared to differentiate the damn polynomial.

* questions on multiple roots often go like this:
p(x) = ax^3 + bx^2 + cx + d has a double root.
a) find what it is
b) hence prove some relationship between the coefficients of p(x)

in this case, find p'(x) to do (a), and sub (a) into p(x) to do (b).

* be prepared to sub roots into the polynomial. this is often very useful. example question? HSC 2002 Q8a

* know relationship between roots and coefficients. in particular, sums and products of roots.

* know your special rules, and how to prove them (e.g. conjugate root thm, conditions for multiple roots)

* know how to factorise symmetrical quartics i.e. Ax^4 + Bx^3 + Cx^2 + Bx + A

* construction of polynomial with given roots:
syllabus proclaims you need to know how to construct P(x) with roots:
- ka, kb, kc
- a+k, b+k, c+k
- 1/a, 1/b, 1/c
- a^2, b^2, c^2.
note that all other questions are simply combinations of the above 4 forms.

* know factorisations:
a^n - b^n = (a-b)(a^(n-1) + a^(n-2)b + a^(n-3)b^2 + ... + b^(n-1))
n is odd:
a^n + b^n = (a+b)(a^(n-1) - a^(n-2)b + a^(n-3)b^2 - ... + b^(n-1))
these are particularly useful in roots-of-unity type questions: e.g. find all roots of z^5 = -1

general tip: want to work faster? look at what answer u're supposed to prove (aka work backwards)

 

[McLake]

To go with spice girl's last point, here is a genral tip: If you get stuck when doing a proof then go to the bottom of the page, write the answer, and work your way up to the bit where you are stuck. This often works well ...

 

[fashionista]

k i got a weeny lil question...but wuts the difference between the roots of a polynomial and the zeros of a polynomial?
thanking u muchly

 

[abdooooo!!!]

there is a difference??? ahh... i thought they were the same

 

[fashionista]

thats wut i thought but my teacher was saying roots are....and zeros are....and i was extremely tired of the sleepy variety so i coincidentally tuned out exactly wen he said the words that go where i put tha dotties.

 

[McLake]

Roots are different to zeros.

A polynomial has a root, an equation has a zero.

eg: the roots of y = x^2 - 1 are +/-1,
the zeros of x^2 - 1 are +/-1.

It's subtle, but different ...

 

[freaking_out]

Originally posted by abdooooo!!!
there is a difference??? ahh... i thought they were the same

well its part of the syllabus to know the difference i think!

 

[abdooooo!!!]

syllabus??? who the hell reads that?

are they gonna give you a question like "explain qualitatively the difference between roots and zeros?"

 

[freaking_out]

Originally posted by abdooooo!!!
...are they gonna give you a question like "explain qualitatively the difference between roots and zeros?"

well u never know these days can u? just look at the science subjects.