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) = x^{3} + x^{2} + x + 1

g(x) = x + 2

q(x) = x^{2} - x + 3

r (x) = -5

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

Here,

L.H.S = x^{3} + x^{2} + x + 1

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

= x^{3} + 2x^{2} – x^{2} – 2x + 3x + 5 – 5

= x^{3} + x^{2} – 2x + 3x + 6 – 5

= x^{3} + x^{2} + 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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