Question

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) .q(x)+ r(x), where degree r (x) = 0.

Answer 1

||✪✪ QUESTION ✪✪||

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) .q(x)+ r(x), where degree r (x) = 0 ?

|| ✰✰ ANSWER ✰✰ ||

Let f(x) = 2x⁴ + 8x³ + 6x² +4x +12

→ q(x) = 2

→ g(x) = x⁴ + 4x³ + 3x² + 2x + 1

→ r(x) = 10

Here, degree r(x) = 0

Now, we know that :--

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)

Dividend = Divisor × Quotient + Remainder

Putting values now,

f(x) = q(x) × g(x) + r(x)

→ 2( x⁴ + 4x³ + 3x² + 2x + 1) + 10

→ 2x⁴ + 8x³ + 6x² +4x + 2 +10

→ 2x⁴ + 8x³ + 6x² +4x +12

Hence, above values of f(x) ,q(x) , g(x) and r(x) satisfy the division algorithm.

Answer 2

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..