Question

If 2 and -2 are two zeroes of the polynomial (x⁴ + x³ – 34x² – 4x + 120), find all the zeroes of the given polynomial.

Answer 1

Let f(x) = x⁴ + x³ – 34x² – 4x + 120 

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

Consequently, (x – 2) (x + 2) = (x² – 4) is a factor of f(x). 

On dividing f(x) by (x2 – 4), 

we get:

f(x) = 0 

⇒ (x² + x – 30) (x² – 4) = 0

⇒ (x² + 6x – 5x – 30) (x – 2) (x + 2) 

⇒ [x(x + 6) – 5(x + 6)] (x – 2) (x + 2) 

⇒ (x – 5) (x + 6) (x – 2) (x + 2) = 0 

⇒ x = 5 or x = -6 or x = 2 or x = - 2 

Hence, 

all the zeroes are 2, -2, 5 and - 6.