Question

Find all the zeroes of the polynomial given below having given numbers as its zeroes.

x⁴ + x³ – 34x² – 4x + 120; 2, – 2.

Answer 1

x⁴ + x³ – 34x² – 4x + 120;2, – 2.

Given zeroes are – 2 and 2

So, (x + 2)and (x – 2) are the factors of x⁴ + x³ – 34x² – 4x + 120

⟹ (x + 2)(x – 2) = x² – 4 is a factor of given polynomial.

Consequently, x² – 4 is also a factor of the given polynomial.

Now, let us divide x⁴ + x³ – 34x² – 4x + 120 by x² – 4

The division process is

Here, quotient = x² + x – 30

= x² + 6x – 5x – 30

= x(x + 6) – 5(x + 6)

= (x + 6)(x – 5)

So, the zeroes are – 6 and 5

Hence, all the zeroes of the given polynomial are – 2 , – 6, 2 and 5.