. 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 r(x) = 0

Best Answer

Solution: Degree of a polynomial is the highest power of the variable in the polynomial.

(i) deg p(x) = deg q(x)

Degree of quotient will be equal to degree of dividend when divisor is constant.

Let us assume the division of 6x2+2x+2 by 2.

Here, p(x) =  6x2+2x+2 by 2.

 

polynomial

 

Thus, the division algorithm is satisfied.

(iii)deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of x3 + 1by x2.

Here,

p(x) = x3 + 1 g(x) = x2

q(x) = x and r(x) = 1

Clearly, the degree of r(x) is 0. Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)x3 + 1

= (x2 ) × x + 1 x3 + 1 = x3 +1

Thus, the division algorithm is satisfied.

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