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Question
What must be added to the polynomial f(x) = x4 + 2x3 − 2x2 + x − 1 so that the resulting polynomial is exactly divisible by x2 + 2x − 3 ?
Solution
We know that,
f(x) = g(x) x q(x) + r(x)
f(x) - r(x) = g(x) x q(x)
f(x) + {- r(x)} = g(x) x q(x)
Clearly , Right hand side is divisible by g(x).
Therefore, Left hand side is also divisible by g(x).Thus, if we add - r(x) to f(x), then the resulting polynomial is divisible by g(x).
Let us now find the remainder when f(x) is divided by g(x).
Hence, we should add - r(x) = x - 2 to f(x) so that the resulting polynomial is divisible by g(x).
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