Question

If 3 and –3 are two zeroes of the polynomial (x⁴ + x³ – 11x² – 9x + 18), find all the zeroes of the given polynomial.

Answer 1

Let x⁴ + x³ – 11x² – 9x + 18

Since 3 and –3 are the zeroes of f(x), it follows that each one of (x + 3) and (x – 3) is a factor of f(x).

Consequently, (x – 3) (x + 3) = (x² – 9) is a factor of f(x).

On dividing f(x) by (x² – 9), we get:   

`x^2 - 9")"overline(x^4 + x^3 - 11x^2 - 9x + 18)"("x^2 + x - 2`
            x⁴             – 9x²
            –               +                          
           x³ – 2x² – 9x + 18
           x³           – 9x
           –              +                           
                        –2x² + 18
                        –2x² + 18
                        +       –                    
                             x                        

f(x) = 0 ⇒ (x² + x – 2) (x² – 9) = 0

⇒ (x² + 2x – x – 2) (x – 3) (x + 3)

⇒ (x – 1) (x + 2) (x – 3) (x + 3) = 0

⇒ x = 1 or x = –2 or x = 3 or x = –3

Hence, all the zeroes are 1, –2, 3 and –3.