Question

5. 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)
(ü) deg g(x) = deg r(x)
(iii) deg r(x)=0 with full steps ​

Answer 1

(i) Let us assume the division of 6x2 + 2x + 2 by 2

Here, p(x) = 6x^2 + 2x + 2

g(x) = 2

q(x) = 3x^2 + x + 1

r(x) = 0

Degree of p(x) and q(x) is same i.e. 2.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

Or, 6x^2 + 2x + 2 = 2× (3x^2 + x + 1)

Hence, 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., 1.

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) Let us assume the division of x^3+ 1 by 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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Answer 2

Step-by-step explanation:

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