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Question

Let p(x) be a polynomial of degree more than 2. When p(x) is divided by x-2, it leaves remainder 1 and when it is divided by x-3, it leaves remainder 3. Find the remainder, when p(x) is divided by (x-2)(x-3).

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Solution

The remainder of division by a degree polynomial will be a degree 1 polynomial, that is, something in the form ax + b. Hence we have, for any polynomial p, there exists a polynomial q and numbers a and b such that:

p(x) = q(x)(x - 3)(x - 2) + ax + b

By remainder theorem, the remainder from division by (x - a) of p is given by p(a). Hence, the remainder from dividing by (x - 2) is p(2) = 1, and similarly p(3) = 3

Hence:

p(2) = q(2)(2 - 3)(2 - 2) + 2a + b = 1
2a + b = 1 ... (1)

and:

p(3) = q(3)(3 - 3)(3 - 2) + 3a + b = 3
3a + b = 3 ... (2)

So now we have a linear system of equations. Subtracting (1) from (2):

a = 2

and hence:

2(2) + b = 1 ... (1)
b = -3

So the remainder is 2x - 3.

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