2004 TMSCA

[gemini9525]

how do you do these?

the least amount of non real zeros of 11x^10+8x^3+x^2-x+2=0 is?

how many five-letter groups can be formed the letters of the word EQUATIONS if three are consonants and two are vowels?

*the answer to the first one is 6 and second one is 480, but neither one makes any sense to me*

[NickR]

For the first one you use Descartes' rule of signs:

First, find possible number of positive(real) roots...

[unparseable or potentially dangerous latex formula]
Now, look and count the number of sign changes...which we determine to be 2.
As per the rule, this means that the number of positive, real roots could either be [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula].

Now, let's find the possible number of negative(real) roots and we find this by using [unparseable or potentially dangerous latex formula] instead of [unparseable or potentially dangerous latex formula] in our function/polynomial.

[unparseable or potentially dangerous latex formula] [unparseable or potentially dangerous latex formula]
Counting the number of sign changes yields us 2 again. So that means we can either have be [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula] negative, real roots.

Since the problem wants the least number of non real zeros or roots, that means we want to maximize the number of real roots. Thus, the max number of real roots is [unparseable or potentially dangerous latex formula] and that leaves us with [unparseable or potentially dangerous latex formula] non real roots.

Now, for the second one:
*edited*
The way we can do this problem is with permutations. There are [unparseable or potentially dangerous latex formula] ways to choose the consonants since there are 4 possible choices and you only need 3. Next, there are [unparseable or potentially dangerous latex formula] ways to choose the vowels since there are 5 possible choices and you only need 2. Thus, the total is [unparseable or potentially dangerous latex formula]

[stupidityismygam]

the second one wants permutations

[NickR]

Are you sure? I thought when the question said 'groups' that the order didn't matter.

[peterliang]

the correct answer is 480, not 40, so yes pr

[NickR]

Hmm, I suppose so, but the word 'group' just threw me off.

[kashhustler]

hmm yeah i think the problem is poorly worded

group implies that order doesn't matter

[gemini9525]

how did u get two sign changes when u check for positive roots?

[gemini9525]

oh i see. nvm.

[AuSmith]

[MichaelGummelt]

I agree with AuSmith, I always hated those questions.