Question

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

2x⁴ + 7x³ – 19x² – 14x + 30;√(2, – √2)

Answer 1

2x⁴ + 7x³ – 19x² – 14x + 30;√(2, – √2)

Given zeroes are √2 and – √2

So, (x – √2)and (x + √2) are the factors of 2x⁴ + 7x³ – 19x² – 14x + 30

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

Consequently, x² – 2 is also a factor of the given polynomial.

Now, let us divide 2x⁴ + 7x³ – 19x² – 14x + 30 by x² – 2

The division process is

Here, quotient = 2x² + 7x – 15

= 2x² + 10x – 3x – 15

= 2x(x + 5) – 3(x + 5)

= (2x – 3)(x + 5)

So, the zeroes are – 5 and 3/2

Hence, all the zeroes of the given polynomial are – 5, – √2, √2 and 3/2.