Question

Give examples of polynomials p(x) q(x) and r(x) which satisfy the division algorithm

Answer 1

Step-by-step explanation:

According to the division algorithm, if p(x) and g(x) are two polynomials with g(x)  0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) x q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

(i)    Degree of quotient will be equal to degree of dividend when divisor is constant.

Let us consider the division of  by 3.

Here, p(x) =   and g(x) = 3

q(x) =   and r(x) = 0

Here, degree of p(x) and q(x) is the same which is 2.

Checking:

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

 

Thus, the division algorithm is satisfied.

(ii)    Let us consider the division of 2x4 + 2x by 2x3,

Here, p(x) = 2x4 + 2x and g(x) = 2x3

q(x) = x and r(x) = 2x

Clearly, the degree of q(x) and r(x) is the same which is 1.

Checking,

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

2x4 + 2x =  (2x3 ) x x  + 2x

2x4 + 2x = 2x4 + 2x

Thus, the division algorithm is satisfied.

(iii)    Degree of remainder will be 0 when remainder obtained on division is a constant.

Let us consider the division of 10x3 + 3 by 5x2.

Here, p(x) = 10x3 + 3 and g(x) = 5x2

q(x) = 2x and r(x) = 3

Clearly, the degree of r(x) is 0.

Checking:

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

10x3 + 3 = (5x2 ) x 2x  +  3

10x3 + 3 = 10x3 + 3

Thus, the division algorithm is satisfied.