Question

Obtain all zeroes of the polynomial f(x)= 2x⁴ – 2x³ – 7x² + 3x + 6, if the two of its zeroes are −√(3/2) and √(3/2).

Answer 1

Given, 

f(x)= 2x⁴ – 2x³ – 7x² + 3x + 6 

Since, two of the zeroes of polynomial are −√(3/2) and √(3/2) so, (x + √(3/2)) and (x –√(3/2)) are factors of f(x). 

⇒ x² – (3/2) is a factor of f(x). Hence, performing division algorithm, we get

⇒ f(x)= (2x² – 2x – 4)( x² – 3/2)

= 2(x² – x – 2)( x² – 3/2) 

So, putting x² – x – 2 = 0 we can get the other 2 zeros. 

⇒ (x – 2)(x + 1) = 0 

∴ x = 2 or -1 

Hence, all the zeros of the polynomial are −√(3/2), -1, √(3/2) and 2.