Unknown polynomial...........

Algebra Level 3

An unknown polynomial yields a remainder of 2 upon division by $x-1$ , and a remainder of 1 upon division by $x-2$ . If this polynomial is divided by $(x-1)$ $(x-2)$ , then the remainder is.

Details and assumptions :

You have to write only the constant if, your answer is $x+1$ then you have to write only 1.

The answer is 3.

1 solution

Mitchell Price
Apr 23, 2014

Our polynomial, let's call it f(x), can essentially be written as such:

$f(x)\quad =\quad g(x)(x-1)(x-2)\quad +\quad remainder$

g(x) is just our quotient when we divide the polynomial by (x-1)(x-2). Now, our remainder has to be of a smaller degree than (x-1)(x-2), so it has to be of degree 1 or 0. It cannot, however, be of degree 0, as a remainder of degree 0 would return the same remainder when we divide it by both (x-1) and (x-2), which does not occur. Therefore, we can write our remainder in the form ax+b. Now, all we need to do is rely on the remainder theorem and plug in our values for x.

When divided by x-1, the remainder is 2, so

$a(1)\quad +\quad b\quad =\quad a\quad +\quad b\quad =\quad 2$

When divided by x-2, the remainder is 1, so

$a(2)\quad +\quad b\quad =\quad 2a\quad +\quad b\quad =\quad 1$

Solving for a and b yields a = -1 and b = 3. This makes our remainder -x + 3

Unknown polynomial...........

The answer is 3.

1 solution

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