Question

It is given that –1 is one of the zeroes of the polynomial `x^3 + 2x^2 – 11x – 12`. Find all the zeroes of the given polynomial.

Answer 1

Let f(x) =` x^3 + 2x^2 – 11x – 12`
Since – 1 is a zero of f(x), (x+1) is a factor of f(x).
On dividing f(x) by (x+1), we get 

 

`f(x) = x^3 + 2x^2 – 11x – 12`
`= (x + 1) (x^2 + x – 12)`
`= (x + 1) {x^2 + 4x – 3x – 12}`
`= (x + 1) {x (x+4) – 3 (x+4)}`
`= (x + 1) (x – 3) (x + 4)`
`∴f(x) = 0 ⇒ (x + 1) (x – 3) (x + 4) = 0`
`⇒ (x + 1) = 0 or (x – 3) = 0 or (x + 4) = 0`
`⇒ x = – 1 or x = 3 or x = – 4`
Thus, all the zeroes are – 1, 3 and – 4.