Question

and-2x+4, respectively. Find g(x).
5. Give examples of polynomials p(x), g(x), g(x) and P(x), which satisfy the division algorithm
and
(1) deg p(x) = deg g(x)
(ii) deg g(x) = deg r(x)
(iii) deg r(x)=0​

Answer 1

Answer:

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iii is the answer..

Answer 2

Step-by-step explanation:

According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula;

Dividend = Divisor×Quotient+Remainder

∴ p(x) = g(x)×q(x)+r(x)

Where r(x) = 0 or degree of r(x)< degree of g(x).

Now let us proof the three given cases as per division algorithm by taking examples for each.

(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.