Question

Find the values of a and b in the polynomial f(x) = 2x³ + ax² + bx + 10, if it is exactly divisible by (x+2) and (2x-1).

Answer 1

(x+2)  ⇒  x = -2 .... (i) 

(2x - 1) ⇒ x = `1/2` .......(ii)

Putting (i) in polynomial, we get 

f(-2) = 2 × (-2) × (-2) × (-2) + a × (-2) × ( -2) + b × (-2) + 10 = 0 

⇒ - 16 + 4a - 2b + 10 = 0

⇒ a = `"b"/2 + 3/2`  .....(iii)

Putting (ii) in polynomial, we get 

`"f"(1/2) = 2 xx (1/2) xx (1/2) xx (1/2) + "a" xx (1/2) xx (1/2) + "b" xx (1/2) + 10 = 0`

`=> 1/4 + "a"/4 + "b"/2 + 10 = 0`

⇒ a = - 2b - 41   ......(iv)

Combining (iii) and (iv), we get, 

`"b"/2 + 3/2 = "a" = -2"b" - 41`

`=> ("b + 3")/2 = - 2 "b" - 41`

⇒  b + 3 = - 4b - 82 

⇒ 5b = - 85 

⇒ b = -17

and a = -7

⇒ a = - 7 , b = -17