Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

Question : Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i)  deg p(x) = deg q(x)

(ii)  deg q(x) = deg r(x)

(iii)  deg q(x) = 0

Solution :

(i)  Let q(x) = \(3x^2 + 2x + 6\),                 Degree of q(x) = 2

p(x) = \(12x^2 + 8x + 24\),                       Degree of p(x) = 2

Here, deg p(x) = deg q(x)

(ii)  Let p(x) = \(x^5 + 2x^4 + 3x^3 + 5x^2 + 2\),

q(x) = \(x^2 + x + 1\),                       Degree of q(x) = 2

g(x) = \(x^3 + x^2 + x + 1\)

r(x) = \(2x^2 – 2x + 1\),                     Degree of r(x) = 2

Here, deg q(x) = deg r(x)

(iii)  Let p(x) = \(2x^4 + 8x^3 + 6x^2 + 4x + 12\),

q(x) = 2,                       Degree of q(x) = 0

g(x) = \(x^4 + 4x^3 + 3x^2 + 2x + 1\)

r(x) = 10

Here, deg q(x) = 0

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