Q. 53.9( 174 Votes )

# Give examples of polynomials and which satisfy the division algorithm and

(i)

(ii)

(iii)

Answer :

Degree of a polynomial is the highest power of the variable in the polynomial. For example if f(x) = x^{3} - 2x^{2 }+ 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)

We need to start with p(x) = q(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) = x^{2} + 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 *x*^{3}*+ x* by *x*^{2},

Here*,*

*p*(*x*) = *x*^{3}*+ x*

*g*(*x*) = *x*^{2}

*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) x*^{3} *+ x*

= (*x*^{2} ) × *x* + *x x*^{3} *+ x = x*^{3} *+ 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 *x*^{3}*+* 1by *x*^{2}.

Here*,*

*p*(*x*) = *x*^{3}*+* 1 *g*(*x*) = *x*^{2}

*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)x*^{3} *+* 1

= (*x*^{2} ) × *x* + 1 *x*^{3} *+* 1 *= x*^{3} *+*1

Thus, the division algorithm is satisfied.

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