# Give examples of polynomials and which satisfy the division algorithm and(i) (ii) (iii) Degree of a polynomial is the highest power of the variable in the polynomial. For example if f(x) = x3 - 2x+ 1, then the degree of this polynomial will be 3.

(i) By division Algorithm : p(x) = g(x) x q(x) + r(x)
It means when P(x) is divided by g(x) then quotient is q(x) and remainder is r(x)
This means that the degree of polynomial p(x) and quotient q(x) is same. This can only happen if the degree of g(x) = 0 i.e p(x) is divided by a constant
Let p(x) = x2 + 1 and g(x) = 2
The, Clearly, Degree of p(x) = Degree of q(x)
2. Checking for division algorithm,

p(x) = g(x) × q(x) + r(x) = Thus, the division algorithm is satisfied.

(ii) Let us assume the division of x3+ x by x2,

Here,

p(x) = x3+ x

g(x) = x2

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

Clearly, the degree of q(x) and r(x) is the same i.e.,
Checking for division algorithm,

p(x) = g(x) × q(x) + r(x) x3 + x

= (x2 ) × x + x x3 + x = x3 + x

Thus, the division algorithm is satisfied.

(iii) Degree of the remainder will be 0 when the 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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