polynomial division - pls help

[Bahnanna]

hi, I am having trouble with this problem:
Suppose that a and b are integers such that x^2 - x - 1 and ax^3 + bx^2 + 1. What is b?
any help you be useful.

 

[Deleted member 4993]

hi, I am having trouble with this problem:
Suppose that a and b are integers such that x^2 - x - 1 and ax^3 + bx^2 + 1. What is b?
any help you be useful.

You don't say "such that what?

Please share your work with us .

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...217#post322217

 

[stapel]

Suppose that a and b are integers such that x^2 - x - 1 and ax^3 + bx^2 + 1....

...such that (this expression) and (that expression) are... what? Equal to something? Related somehow? Something else? ;-)

 

[Bahnanna]

so i read the 'read before you post' and I am sorry for any trouble i might have caused.
I am in Yr11 High level math and out of the problem that I listed above, i have been told to use long division which is where i am having trouble. I have tried it a number of times and am unable to get 0 as a remainder which it must be because x^2 -x-1 is a factor, therefore the remainder must 0.
If you could give me any help that would be wonderful

 

[HallsofIvy]

You still haven't told us what the problem is! You are given twp polynomials "such that" what? Do you not understand that "such that" must be followed by some property? Please give the full, exact statement of the problem.

 

[Bahnanna]

So:
x^2-x-1 is a factor of ax^3 + bx^2 + 1
and a and b are integers
and the question is asking us to find what b is. sorry for the bad wording

 

[Deleted member 4993]

So:
x^2-x-1 is a factor of ax^3 + bx^2 + 1
and a and b are integers
and the question is asking us to find what b is. sorry for the bad wording

Hint:

ax^3 + bx^2 + 1 = ax(x^2 - x - 1) + (a+b)(x^2 - x - 1) + (2a + b)x + (a+b-1)

 

[Bahnanna]

So i tried using that equation but I am just not quite sure how to apply it in the context of the, to find b

 

[DrPhil]

So:
x^2-x-1 is a factor of ax^3 + bx^2 + 1
and a and b are integers
and the question is asking us to find what b is. sorry for the bad wording

You said you tried the long division - that should have worked. The remainder I got was of the form Cx + D, where C and D are functions of a and b. Since x may have any value, the remainder is identically 0 ONLY if both C and D are 0. That gives two equations in the two unknowns, a and b.

Show us what you got by dividing. Can you find a and b using this hint?

 

[Bahnanna]

So the remainder would be

(2a+b)x +(a+b-1) which would be =0

Then using the hint above we know that :

(2a+b) =0
(a+b-1)=0

then using simultaneous equations I got that:
a=-1
b=2

is that correct??

 

[DrPhil]

So the remainder would be

(2a+b)x +(a+b-1) which would be =0

Then using the hint above we know that :

(2a+b) =0
(a+b-1)=0

then using simultaneous equations I got that:
a=-1
b=2

is that correct??

I hope so - that is what I got too!

 

[Deleted member 4993]

So the remainder would be

(2a+b)x +(a+b-1) which would be =0

Then using the hint above we know that :

(2a+b) =0
(a+b-1)=0

then using simultaneous equations I got that:
a=-1
b=2

is that correct??

Use those "back" in your original equations - and find out.