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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)