Question

An unknown polynomial is divided by x-1 to give a remainder 2 and with x-2 to give a remainder 1.Then the remainder on dividing it by (x-1)(x-2)

Answer 1

I assume you meant to write "if an unknown polynomial..." in which case, let us denote the unknown polynomial by f(x).

 

We can encode the two conditions on the polynomial as follows:

 

f(x) = q1(x)⋅(x-1) + 2

f(x) = q2(x)⋅(x-2) + 1

 

Now, we multiply the first equation by (x-2), and the second by (x-1) to obtain

 

(x-2)f(x) = q1(x)⋅(x-1)(x-2) + 2(x-2)

(x-1)f(x) = q2(x)⋅(x-2)(x-1) + 1(x-1)

 

and upon subtracting the first of these resulting equations from the second, we obtain

 

f(x) = (q2(x) - q1(x))⋅(x-1)(x-2) -x+3

 

From this last equation, we see that the remainder upon division by (x-1)(x-2) must be -x+3.

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