Math CentralQuandaries & Queries


Question from Asterdine, a student:

When a polynomial f(x) is divided by x+1 and x+2, the remainders are 3 and 7 respectively. Find the remainder when f(x) is divided by (x+2)(x+1).


Since (x + 2)(x + 1) is a quadratic polynomial the remainder after dividing f(x) by (x + 2)(x + 1) must be a linear polynomial which means that there is a polynomial g(x) and numbers a and b so that

f(x) = g(x) (x + 2)(x + 1) + ax + b.

Since division of f(x) my (x + 1) leaves a remainder of 3 the remainder theorem says that

f(-1) = 7.

Substitute x = -1 into f(x) = g(x) (x + 2)(x + 1) + ax + b and this will give you a linear equation in the variables a and b. Use the fact that division of f(x) my (x + 2) leaves a remainder of 7 to find a second linear equation in a and b. Solve for a and b.


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