Question

f(x) divided by x-2 gives remainder 3 and whenf(x) is divided by x-3 gives remainder 2.what will be the remainder when f(x) divided by (x-2)(x-3)

Answer 1

Answer:

-x+5

Step-by-step explanation:

When divided by x-a, the remainder left is the same as the value of f(a).  This is the Remainder Theorem.

Since f(x) leaves remainder 3 when divided by x-2, we know f(2) = 3.

Since f(x) leaves remainder 2 when divided by x-3, we know f(3) = 2.

When dividing by (x-3)(x-2), we get

f(x) = (x-3)(x-2)(some polynomial) + Px + Q

where Px+Q is the remainder that we seek.

Putting x = 2, we get 2P + Q = f(2) = 3.

Putting x = 3, we get 3P + Q = f(3) = 2.

Solving these two equations for P and Q, we have P = -1 and Q = 5.

The remainder is -x + 5.