Question

The remainder when x^5+kx^2 is divided by (x-1)(x-2)(x-3) contains no terms in x^2. Find k

Answer 1

(x – 1) (x – 2) (x – 3) = x^3 – 6x^2 + 11x – 6
Let the quotient and remainder when
x^5 + kx^2 is divided by (x – 1) (x – 2) (x – 3) be
(ax^2 + bx + c) and dx + e
Now
x^5 + kx^2 = (x^3 – 6x^2 + 11x – 6) (ax^2 + bx + c) + dx + e
Comparing the coefficients of x^5, x^4, x^3 and x^2we get
a = 1
b – 6 a = 0 ⇒ b = 6
c – 6 b + 11 a = 0 ⇒ c = 25
– 6 c + 11 b – 6 a = k
⇒k = –90
Hence, the value of k is –90