Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) .q(x)+ r(x), where degree r (x) = 0.

Let f(x) = x3 + x2 + x + 1


g(x) = x + 2


q(x) = x2 - x + 3


r (x) = -5


Now, f(x) = g(x) .q(x)+ r(x)


Here,


L.H.S = x3 + x2 + x + 1


And, R.H.S = (x + 2) × (x2 - x + 3) + (-5)


= x3 + 2x2 – x2 – 2x + 3x + 5 – 5


= x3 + x2 – 2x + 3x + 6 – 5


= x3 + x2 + x + 1


= R.H.S


∴ the given set of polynomials satisfies the equation: f(x) = g(x) .q(x)+ r(x)


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