We know that, Show
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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The remainder is 2; when x⁴+ x³-2x²+ x + 1 is divided by x-1. Given: Two polynomials. x⁴+ x³-2x²+ x + 1.
What must be subtracted from 4x⁴ 2x³ 6x² 2x 6 so that the result is exactly divisible by 2x² x 1?(ii) <br> On solving (i) and (ii) , we get a= 1 and b = 5 . <br> Hence , the required expression to be subtracted is (x+5).
What must be subtracted from x3 6x2 15x 80 so that the result is exactly divisible by x2 x 12?Hence, x 3 - 6 x 2 - 15 x + 80 will be divisible by x 2 + x - 12 , if 4 x − 4 is subtracted from it.
What must be subtracted from X 2x3 2x² 4x 6 so that the result is exactly divisible by X² 2x 3 )?Thus, the remainder (2x + 9) must be subtracted from (x4 + 2x3 – 2x2 + 4x + 6) so that the result is exactly divisible by (x2 + 2x – 3).
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