Question

Find the values of a and b when the factors of the polynomial f(x)= ax³ + bx² + x a are (x+3) and (2x-1). Factorize the polynomial completely.

Answer 1

(x+3) ⇒ x = -3 .....(i)

(2x - 1) ⇒ x = `1/2` .....(ii)

Putting (i) in polynomial, we get 

f (-3) = a × (-3) × (-3) × (-3) + b × (-3) × (-3) + ( -3) - a = 0 

⇒ 27 a + 9 b - 3 - a = 0

⇒ `"a" = "9b"/28 - 3 /28`  ........(iii)

Putting (ii) in polynomial, we get 

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

`=> "a"/8 + "b"/4 + 1/2 - "a" = 0`

`=> "b" = "7a"/2 - 2`  .........(iv)

Combining (iii) and (iv), we get, 

`"a" = 9/28 xx ("7a"/2 - 2) - 3/28`

⇒  56 a = 63 a - 42 

⇒ a= 6  

⇒ b = `(7 xx 6)/2 - 2 = 21 - 2 = 19`

a = 6 , b = 19

Putting these values in polynomial, we get 

f(x) = 6x³ + 19x² + x - 6 

Hence, equation becomes (x + 3) (2x - 1)(3x + 2) = 0