roots/zero (1 Viewer)

[hasterz]

what is the difference between a root and a zero? and btw its not the same thing, the syllabus actually says u need to differentiate between the 2

 

[lucifel]

i think zero just refers to the number, where as roots is actually like (x-2) or something, i am not entirely sure.

 

[Trev]

I'm not too sure, but I have never heard the term 'complex zero', so maybe root refers to all thingys over the complex field, but zero's are only for real values....
[correct me if I am wrong, which is most likely the case]

 

[KFunk]

I'm pretty sure that they're the same thing i.e
For the polynomial (x-a)(x-b)=0
It has roots: x=a and x=b
It has zeros: x=a and x=b
It has factors: x-a and x-b

EDIT: http://mathworld.wolfram.com/Root.html (I swear we had a drawn out discussion of this in class dude )

 

[Slidey]

Zero: "An argument at which the value of a function vanishes."
That is: for f(x), f(a)=0, a is a zero
Root:
"A number that reduces a polynomial equation in one variable to an identity when it is substituted for the variable.
A number at which a polynomial has the value zero."
That is: for f(x), f(a)=0, a is a zero

KFunk is correct.

 

[Will Hunting]

They are indistinguishable in principle, however, they are distinguishable in practice.

In other words, if a question asked you only to find the roots, without mention of the zeroes, or vice versa, there ARE different ways to express your answer.

For roots, you would write:

The roots are, x = a, b, ..., n for a p'nomial with n distinct roots

For zeroes, you would write, simply:

The zeroes are a, b, c, ..., n (no x =)

So, it can be seen that roots can be considered values of x giving P(x) = 0, whereas zeroes are just values.

Other than this, though, they are identical in function, purpose and idea.

 

[KFunk]

They are indistinguishable in principle, however, they are distinguishable in practice.

In other words, if a question asked you only to find the roots, without mention of the zeroes, or vice versa, there ARE different ways to express your answer.

For roots, you would write:

The roots are, x = a, b, ..., n for a p'nomial with n distinct roots

For zeroes, you would write, simply:

The zeroes are a, b, c, ..., n (no x =)

So, it can be seen that roots can be considered values of x giving P(x) = 0, whereas zeroes are just values.

Other than this, though, they are identical in function, purpose and idea.

That seems to be more a matter of grammar and semantics (and perhaps your personal method of communication/notation) than of mathematics. That kind of 'distinction' just muddies the waters IMO.

 

[Will Hunting]

That seems to be more a matter of grammar and semantics (and perhaps your personal method of communication/notation) than of mathematics

Absolutely, dude, and I was in no way oblivious to this when I made my addition to the thread. However, you are wrong to dissuade consideration of it at all. How does its being a syntactical issue milden, in any way, its importance to mathematics? Trivial as it may be, it too has its role to play and, as such, should not be overlooked.

By the way, nah, it's not my personal method. It was something taught me by my teacher, who felt very strongly about its significance.

That kind of 'distinction' just muddies the waters IMO

Your call, man, but, again, this doesn't attenuate its importance.

 

[KFunk]

Absolutely, dude, and I was in no way oblivious to this when I made my addition to the thread. However, you are wrong to dissuade consideration of it at all. How does its being a syntactical issue milden, in any way, its importance to mathematics? Trivial as it may be, it too has its role to play and, as such, should not be overlooked.

Yeah, I'll pay that (EDIT: but only just ). It was mainly the statement "So, it can be seen that roots can be considered values of x giving P(x) = 0, whereas zeroes are just values." which I kinda objected to. Just be careful with gospel statements when people are unsure of concepts. [as you can probably tell I don't agree with the distinction you made ]

 

[Slidey]

For roots, you would write:

The roots are, x = a, b, ..., n for a p'nomial with n distinct roots

For zeroes, you would write, simply:

The zeroes are a, b, c, ..., n (no x =)

So, it can be seen that roots can be considered values of x giving P(x) = 0, whereas zeroes are just values.

This is actually not really the correct distinction.

A root is a value of x which satisfies f(x)=0.

A zero is a value which makes f(x) vanish.

For example:

(x-3)(x+2)=y
Has zeros x=3 and x=-2, but has roots x=3 and x=-2 iff y=0, whereas:
(x-3)(x+2)=0
Has zeros x=3 and x=-2 and has roots x=3 and x=-2.

 

[dawso]

yeah, the way i take it, if it just factorises for y, it is a zero, whereas if it is a function equal to zero and then u take the value that will giv it a root on the axis, then it is a root, kinda like, the opposite of wat it says.... hard 2 explain i suppose...

 

[Slidey]

On the topic of roots and zeros, I highly recommend all 4u students (and 3u, in fact) learn Synthetic Division. There's an excellent page here which explains how to do it:

http://www.purplemath.com/modules/synthdiv.htm

It's much faster than long division and there's less chance of error.

 

[rama_v]

Wow slide, thats very interesting...but will they allow it in the hsc? Or do u only reccomend it for testing answers, or for integration questions where u have to split up the fraction by division?

 

[Slidey]

My teacher said he used to teach it. Also, there's no reason why they wouldn't.

It's a valid method. Might want to write something before-hand like "by synthetic division", though.

But yeah it's also a great way to test. A lot of people use the remainder theorem P(a)=0, then (x-a) is a factor. But it takes just as long to use synthetic division with one bonus: you've actually found the other factor of the polynomial.