When x3 - 2x2 + ax = b is divided by x2 - 2x - 3, the remainder is

1.	When x3 - 2x2 + ax = b is divided by x2 - 2x - 3, the remainder is x-6. The value of a and b respectively A. -2, -6 B. 2 and -6 C. -2 and 6 D. 2 and 6
Answer» Let p(x) = x³ - 2(x²) + ax - b q (x) = x² - 2x - 3 r (x) = x - 6 Therefore, f(x) = p (x) – r (x) f(x) = x³ - 2x² + ax - b - x - 6 = x³- 2x² + (a - 1) x - (b - 6) q(x) = x² - 2x - 3 = (x + 1) (x - 3) Thus, (x + 1) and (x - 3) are factor of f (x) a + b = 4 f (3) = 0 3³ - 2 (3)² + (a-1) 3 - b + 6 = 0 12 + 3a - b = 0 a = - 2, b = 6

When x3 - 2x2 + ax = b is divided by x2 - 2x - 3, the remainder is x-6. The value of a and b respectively A. -2, -6 B. 2 and -6 C. -2 and 6 D. 2 and 6

Answer»

Let p(x) = x³ - 2(x²) + ax - b

q (x) = x² - 2x - 3

r (x) = x - 6

Therefore,

f(x) = p (x) – r (x)

f(x) = x³ - 2x² + ax - b - x - 6

= x³- 2x² + (a - 1) x - (b - 6)

q(x) = x² - 2x - 3

= (x + 1) (x - 3)

Thus,

(x + 1) and (x - 3) are factor of f (x)

a + b = 4

f (3) = 0

3³ - 2 (3)² + (a-1) 3 - b + 6 = 0

12 + 3a - b = 0

a = - 2, b = 6