Problem with a product of 2 remainders (polynomials)

AI Thread Summary
The discussion revolves around finding the remainder of a polynomial P(x) when divided by (x-2)(x+3), given that P(2)=10 and P(-3)=5. Participants utilize the remainder theorem, establishing that the remainder V(x) must be of the form ax+b. By substituting the known values into the equations V(2)=10 and V(-3)=5, they set up a system of equations to solve for a and b. The conversation emphasizes the importance of correctly applying polynomial division and the remainder theorem to derive the solution. Ultimately, the solution involves solving the linear equations derived from the remainders.
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Homework Statement


[/B]
Polynomial P(x) when divided by (x-2) gives a remainder of 10. Same polynomial when divided by (x+3) gives a remainder of 5. Find the remainder the polynomial gives when divided by (x-2)(x+3).

2. Homework Equations

Polynomial division, remainder theorem

The Attempt at a Solution


From the problem and the remainder theorem I got 2 equations:
1) P(2)=10 and 2) P(-3)=5
More generally, we get this: P(x)=(x-2)D(x)+10=(x+3)F(x)+5 , where D(x), F(x) some polynomials one rank lower than P(x).

Tried multiplicating, dividing and adding the equations in different ways but couldn't get a useful relation, namely P(x)=(x-2)(x+3)G(x)+ V(x), where V(x) is the wanted remainder. Any thoughts?
 
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another_dude said:

Homework Statement


[/B]
Polynomial P(x) when divided by (x-2) gives a remainder of 10. Same polynomial when divided by (x+3) gives a remainder of 5. Find the remainder the polynomial gives when divided by (x-2)(x+3).

2. Homework Equations

Polynomial division, remainder theorem

The Attempt at a Solution


From the problem and the remainder theorem I got 2 equations:
1) P(2)=10 and 2) P(-3)=5
More generally, we get this: P(x)=(x-2)D(x)+10=(x+3)F(x)+5 , where D(x), F(x) some polynomials one rank lower than P(x).

Tried multiplicating, dividing and adding the equations in different ways but couldn't get a useful relation, namely P(x)=(x-2)(x+3)G(x)+ V(x), where V(x) is the wanted remainder. Any thoughts?
Do you know the Chinese remainder theorem?
 
Never heard of it before.
 
another_dude said:
Tried multiplicating, dividing and adding the equations in different ways but couldn't get a useful relation, namely P(x)=(x-2)(x+3)G(x)+ V(x), where V(x) is the wanted remainder. Any thoughts?

##V(x)## must have the form ##ax+b##, right? What are ##V(2)## and ##V(-3)##?
 
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Oh right, didn't think of that actually. Well, from the remainder theorem we get 1) V(2)=P(2)=10 2) V(-3)=P(-3)=5 . Then you solve the system for a and b. Thanks!
 
I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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