Question

If 1 and –2 are two zeroes of the polynomial (x³ – 4x² – 7x + 10), find its third zero.

Answer 1

Let f(x) = x³ – 4x² – 7x + 10

Since 1 and –2 are the zeroes of f(x), it follows that each one of (x – 1) and (x + 2) is a factor of f(x).

Consequently, (x – 1) (x + 2) = (x² + x – 2) is a factor of f(x).

On dividing f(x) by (x² + x – 2), we get: 

`x^2 + x - 2")"overline(x^3 - 4x^2 - 7x + 10)"("x - 5`
                    x³ +    x² – 2x
                    –   –        +                 
                  –5x² – 5x + 10
                  –5x² – 5x + 10
                   +       +        –              
                          X                          
 f(x) = 0 ⇒ (x² + x – 2) (x – 5) = 0

⇒ (x – 1) (x + 2) (x – 5) = 0

⇒ x = 1 or x = –2 or x = 5

Hence, the third zero is 5.