A polynomial f(x) = x4 – 11x3 + 31x2 – 46x + 20 is defined. When it is divided by x2 – 3x + n, the remainder left is q. Find the product of n and q.

A polynomial f(x) = x⁴ – 11x³ + 31x² – 46x + 20 is defined. When it is divided by x² – 3x + n, the remainder left is q. Find the product of n and q.

A 20

B 30

C 40

D 50

E 100

Correct Answer: Option D

Let the quotient after division be x² + ax + b.

⇒ (x² – 3x + n)( x² + ax + b) + q = x⁴ – 11x³ + 31x² – 46x + 20

⇒ x⁴ + (a – 3)x³ + (n – 3a + b)x² + (an – 3b)x + bn + q = x⁴ – 11x³ + 31x² – 46x + 20

Comparing, we get a = -8, n – 3a + b = 31, an – 3b = -46, bn + q = 20.

Put value of a, we get n + b = 7, 8n + 3b = 46

Solving, we get n = 5, b = 2.

Now, bn + q = 20

⇒ q = 20 – 5× 2 = 10

∴ Product of n and q will be 50.