Question

Give examples of polynomials f(x),g(x),q(x)and r(x) which satisfy division Algorithm and (i)degree r(x)=0 (ii)degree q(x) =degree r(x)=1​

Answer 1

hope this helps u

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