Question 5

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

Answer

we know that

(i) Deg P(x) = deg g (x)

The degree of dividend or quotient can be equal, only if the divisor is a constant (degree 0)

Then, let p(x) = 3x^{2 }– 6x + 5

Let g(x) = 3

Therefore, q(x) = x^{2} – 2x +1 and r(x) = 2

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

Let p(x) = x^{2} + 1

Let g(x) = x

Therefore, q(x) = x + 1 and r(x) = 0

Here, we 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 zero, only if the remainder left after division algorithm is Constant.

Let p(x) = x^{2} + 1

Let g(x) = x

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

Here we can see the degree of remainder is zero.

Hence division algorithm is satisfied here.

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