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Question
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.
Solution
Using division algorithm, we have
`f(x)= g(x)xxq(x)+ r(x)`
`x^5- 4x^3 + x^2+ 3x +1 = (x^3 - 3x +1)(x^2 -1)+ 2`
`x^5 - 4x^3+ x^2 + 3x + 1= x^5 - 3x^2 - x^3 + 3x -1 +2`
`x^5 -4x^3 + x^2 +3x +1 = x^5 - 4x^3 + x^2 + 3x +1`
`x^5 - 4x^3 + x^2 + 3x +1 = x^5 - 4x^3 + x^2 + 3x + 1`
Hence an example for polynomial `f(x)`, g(x), q(x) and r (x) satisfying `f(x)= g(x)xxq(x)+ r (x)` are
`f(x)= x^5 - 4 x^3 + x^2 + 3x +1 `
`g(x)= (x^2 - 3x + 1)`
`q(x)= (x^2-1)`
`r(x)= 2`
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