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Question

What must be subtracted from 6x4+13x3+13x2+30x+20, so that the resulting polynomial is exactly divisible by 3x2+2x+5?

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Solution

p(x)=g(x)×q(x)+r(x) [Division algorithm for polynomials]
p(x)r(x)=g(x)×q(x)
It is clear the RHS of the above equation is divisible by g(x). i.e., the divisor.
LHS is also divisible by the divisor.
Therefore, if we subtract remainder r(x) from dividend p(x), then it will be exactly divisible by the divisor g(x)]
Let us divide, 6x4+13x3+13x2+30x+20 by 3x2+2x+5
we get, quotient q(x)=2x2+3x1 remainder r(x)=17x+25
If we subtract 17x+25 from 6x4+13x3+13x2+30x+20, it will be exactly divisible by (3x2+2x+5).
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