Math Is Fun Forum

Carly

Member

Registered: 2006-09-18

Posts: 2

Finding A polynomial

P(x) is divisible by (x+1) and (x-1) , find P(x) if P(2) = 9.

I determined the answer by trial and error and P(x) turned out to be x^3 + x^2 - x - 1 , but is there any systematic method to apply here if the polynomial was more complex ? How can I really solve this algebraicly ?

TIA

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Finding A polynomial

(x+1) | P(x) and (x-1) | P(x).

So:

P(x) = (x+1)(x-1)Q(x) = (x^2-1)Q(x).

P(2) = (3)Q(2) = 9, so Q(2) = 3.

Q(x) = x+1 statifies this.

P(x) = (x+1)(x-1)(x+1) = (x^2-1)(x+1) = x^3 + x^2 - x - 1

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Finding A polynomial

Note that there are multiple solutions:

Q(x) = 3, then:

P(x) = (x^2-1)(3) = 3x^2 - 3

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Carly

Member

Registered: 2006-09-18

Posts: 2

Re: Finding A polynomial

Thanks a lot Ricky!