Remainder Theorem Thinking Question

[fakecop]

Homework Statement

When a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x-1)(x+2)

Homework Equations

The Attempt at a Solution

had the polynomial been a real number, this problem would have been handled with modular mathematics (I think it's called the chinese remainder theorem?) But it isn't-it's polynomial division.

I began by stating that P(-2)=-19 and P(1)=2.

But then I got stuck-there is no way to finding any of the coefficients, because I only have the remainder for two values of x, and the degree of the polynomial isn't specified!

For example, if we assume the polynomial to be cubic. Then we have ax^3+bx^2+cx+d = R(x). I would then substitute the values for x and the remainders for R(x). But there are four unknowns-a, b, c, and d. I can't even solve this system by assuming that the polynomial is cubic, let alone not even knowing the degree.

Now I have come up with some polynomial expressions, such as x^3+4x-3 or 2x^3+x-1 which satisfy the criteria given in the question. Dividing by x^2+x-2, I found the remainder to be 7x-5. (Which is the answer in the book)

But how do I prove that for the general case? (Note that the degree of the polynomial isn't even specified!) Please help.

 

[Dick]

Homework Statement

When a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x-1)(x+2)

Homework Equations

The Attempt at a Solution

had the polynomial been a real number, this problem would have been handled with modular mathematics (I think it's called the chinese remainder theorem?) But it isn't-it's polynomial division.

I began by stating that P(-2)=-19 and P(1)=2.

But then I got stuck-there is no way to finding any of the coefficients, because I only have the remainder for two values of x, and the degree of the polynomial isn't specified!

For example, if we assume the polynomial to be cubic. Then we have ax^3+bx^2+cx+d = R(x). I would then substitute the values for x and the remainders for R(x). But there are four unknowns-a, b, c, and d. I can't even solve this system by assuming that the polynomial is cubic, let alone not even knowing the degree.

Now I have come up with some polynomial expressions, such as x^3+4x-3 or 2x^3+x-1 which satisfy the criteria given in the question. Dividing by x^2+x-2, I found the remainder to be 7x-5. (Which is the answer in the book)

But how do I prove that for the general case? (Note that the degree of the polynomial isn't even specified!) Please help.

Call your polynomial P(x). The remainder when you divide P(x) by (x-1)(x+2) is going to be some linear polynomial ax+b. So you have P(x)=(x-1)f(x)+2, P(x)=(x+2)g(x)-19 and P(x)=(x-1)(x+2)h(x)+ax+b. Put x=1 and x=(-2) into those and see if you can get two equations to solve for a and b.

 

[fakecop]

wow I spent a week thinking about that question and you just helped me solve it in a few minutes.

So I get a+b=2 and -2a+b=-19.
solving, we get a=7 and b=-5. R(x)=7x-5.

So happy to see a simple solution, Thank you!

 

[Dick]

wow I spent a week thinking about that question and you just helped me solve it in a few minutes.

So I get a+b=2 and -2a+b=-19.
solving, we get a=7 and b=-5. R(x)=7x-5.

So happy to see a simple solution, Thank you!

You're welcome. Actually, coming up with sample polynomials that satisfied the premise and working out the remainder was pretty resourceful. That's a good way to attack a problem.