Question 5:

Give examples of polynomial 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:

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) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x)

Degree of a polynomial is the highest power of the variable in the polynomial.

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

Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).

Let us assume the division 6x² + 2x + 2 by 2.

Here, p(x) = 6x² + 2x + 2

g(x) = 2

q(x) = 3x² + x + 1 and r(x) = 0

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

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

6x² + 2x + 2 = 2 (3x² + x + 1)

= 6x² + 2x + 2

Thus, the division algorithm is satisfied.

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

Let us assume the division of x³ + x by x²,

Here, p(x) = x³ + x

g(x) = x²

^{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)

x³ + x = (x² ) × x + x
x³ + x = x³ + x

Thus, the division algorithm is satisfied.

(iii) deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of x³ + 1by x².

Here, p(x) = x³ + 1

g(x) = x²

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)

x³ + 1 = (x² ) × x + 1
x³ + 1 = x³ + 1
Thus, the division algorithm is satisfied.

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