Question

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 1

(i) Let us assume the division of 6x2 + 2x + 2 by 2
Here, p(x) = 6x2 + 2x + 2
g(x) = 2
q(x) = 3x2 + 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, 6x2 + 2x + 2 = 2x (3x2 + x + 1)
Hence, division algorithm is satisfied.


(ii) 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) Let us assume the division of x3+ 1 by 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.

Answer 2

according to EUCLID division lemma :
a = bq + r where 0 ≤ r < b

hence, Let P(x), g(x) , q(x), and r(x) satisfy EUCLID division lemma .
then, P(x) = g(x) × Q(x) + r(x)
where, 0≤ r(x) < g(x)

(i) deg.P(x) = deg.g(x)
Let P(x) = x⁴
g(x) = x⁴ -x
Q(x) = 1
r(x) = x
now,
x⁴ = (x⁴ -x) × 1 + x satisfied //

(iii) degq(x) = degr(x)
Let q(x) = x -1
r(x) = x
g(x) = x²
P(x) = x³ - x² + x
now,
x³ - x² + x = x²(x -1) + x , satisfied //


(iii) deg r(x) = 0
r(x) = 2
g(x) = x +2
q(x) = x³
P(x) = x⁴ + 2x³ + 2
now,
x⁴ + 2x³ + 2 = (x +2)x³ + 2, satisfied//