Question

give examples of polynomials p(x),g(x),q(x) and r(x),which satisfy the division algorithm and (1) deg p(x) =deg q(x) (2) deg q(x)=deg r(x) (3) deg r(x)=0​

Answer 1

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

Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.

Let us take an example, 3x2+3x+3 is a polynomial to be divided by 3.

So, (3x2+3x+3)/3 = x2+x+1 = q(x)

Thus, you can see, the degree of quotient is equal to the degree of dividend.

Hence, division algorithm is satisfied here.

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

Let us take an example , p(x)=x2+x is a polynomial to be divided by g(x)=x.

So, (x2+x)/x = x+1 = q(x)

Also, remainder, r(x) = 0

Thus, you can see, the degree of quotient is equal to the degree of remainder.

Hence, division algorithm is satisfied here.

(iii) deg r(x) = 0

The degree of remainder is 0 only when the remainder left after division algorithm is constant.

Let us take an example, p(x) = x2+1 is a polynomial to be divided by g(x)=x.

So,( x2+1)/x= x=q(x)

And r(x)=1

Clearly, the degree of remainder here is 0.

Hence, division algorithm is satisfied here.