Question

Find the values of a and b when the polynomial f(x)= ax³ + 3x² +bx -3 is exactly divisible by (2x+3) and leaves a remainder -3 when divided by (x+2). 

Answer 1

(2x +3) ⇒ x = `-3/2` .....(i)

(x + 2) ⇒ x = - 2  ...(ii)

putting (i) in polynomial , we get 

`"f"(-3/2) = "a" xx (-3/2) xx (-3/2) xx (-3/2) + 3 xx (-3/2) xx  (-3/2) + "b"  xx (-3/2) - 3 = 0`

- 27 a + 54 - 12 b - 24 = 0

⇒ 27 a = -12 b + 30 ....(iii)

Putting (ii) in polynomial, and remainder is -3 we get 

f(-2) = a × (-2) × (-2) × (-2) + 3 × (-2) × (-2) + b× (-2) - 3 = -3 

b = 6 - 4a ..... (iv)

Combining (iii) and (iv), we get, 

27a = -12 ×(6 - 4a) + 30 

⇒ 27a= -72 + 48a + 30, 

⇒ a=2, b= 6-4x2 = -2 

a= 2, b= -2