Math Is Fun Forum

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Algebra Problem

Part 1:

Let

and

be polynomials.

Suppose

for exactly three values of

: namely,

and

.

Suppose

for exactly five values of

: namely,

and

.

Is it necessarily true that

is divisible by

? If so, carefully explain why. If not, give an example where

is not divisible by

.

Part 2:

Generalize: for arbitrary polynomials

and

, what do we need to know about the zeroes (including complex zeroes) of

and

to infer that

is divisible by

?

(If your answer to Part 1 was "yes", then stating the generalization should be straightforward; try to prove it. If your answer to Part 1 was "no", then try to salvage the idea by imposing extra conditions as needed.)

Can you please help? And if you do, can I have an explanation for how to do it? Thanks.

Last edited by Enshrouded_ (2016-04-13 11:41:26)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problem

Hint: Do you see the factors of those two polys? Use that to show that f(x) will always divide g(x).

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Algebra Problem

I don't get how to see the factors. I know I need a polynomial q(x) so that g(x) = f(x) * q(x), but I don't know how to prove it exists.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problem

f(x) looks like this f (x) = b (x + 3) (x - 4) (x - 8)

g(x) looks like this g (x) = a (x + 5) (x + 3) (x - 2) (x - 4) (x - 8)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Algebra Problem

Oh I see. Thanks

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Algebra Problem

Wait but how do we know that b will divide a(x+5)(x-2)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problem

I think b and a are constants.

with a remainder of 0.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Algebra Problem

Wait, it's false because we could have say f(x)=a(x+3)^5(x-4)(x-8) and it wouldn't work. How do you show a way to determine polynomial divisibility from roots?
(The second part)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problem

f(x)=a(x+3)^5(x-4)(x-8)

You can not pick that because it does not have exactly 3 zeroes.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Problem

Complex roots must be in pairs

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(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}