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

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

We know the formula,

Dividend = Divisor x quotient + Remainder

p(x) = g(x) * q(x) + r(x)

So here, the degree of the quotient will be equal to the degree of dividend when the divisor is constant.

Let us assume the division of 4x2 by 2.

Here, p(x) = 4x2

g(x)=2

q(x)= 2x2

r(x)=0

The degree of p(x) and q(x) is the same, i.e., 2.

Checking for a division algorithm,

p(x) = g(x) * q(x) + r(x)

4x2 } = 2(2x2 )

Hence, the division algorithm is satisfied.

(ii) deg q(x) = deg r(x)

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

The degree of q(x) and r(x) is the same, i.e. 1.

Checking for a division algorithm,

p(x)= g(x) * q(x) + r(x)

x3 + x = x2 * x + x

x3 + x = x3 + x

Hence, the division algorithm is satisfied.

(iii) deg r(x) = 0

The degree of the remainder will be 0 when the remainder comes to a constant.

Let us assume the division of x4 + 1 by x3

Here, p(x) = x4 + 1

g(x) = x3

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

Degree of r(x) is 0.

Checking for a division algorithm,

p(x) = g(x) * q(x) + r(x)

x4 + 1 = x* x + 1

x+ 1 = x4 + 1

Hence, the division algorithm is satisfied.

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