Question

Q-1. what must be added to f(x)= 6x4 + 8x3 + 18x2 + 20x + 5 so that the resulting polynomial is divisible by g(x) = 3x2 + 2x +1

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Answer 1

Given: The polynomial f(x)= 6x4 + 8x3 + 18x2 + 20x + 5

To find: The polynomial to be added in f(x) so that the resulting polynomial is divisible by g(x) = 3x2 + 2x +1

Solution:

• Now we have given f(x) = 6x4 + 8x3 + 18x2 + 20x + 5

• We know the division algorithm, which is:

           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)

• So RHS is divisible by g(x) so LHS is also divisible by g(x).

• So if we add -r(x) to f(x) then resulting polynomial is divisible by g(x).

• So after long division method, we get the final remainder as:

           280x / 9 - 11 / 9

          = 280x - 11 / 9

Answer:

            So here we should add - r(x), that is -(280x - 11 / 9) so that the resulting polynomial is divisible by g(x) = 3x2 + 2x +1.