Question

Obtain all zeroes of the polynomial f(x) = x⁴ – 3x³ – x² + 9x – 6, if the two of its zeroes are −√3 and √3.

Answer 1

f(x) = x⁴ – 3x³ – x2 + 9x – 6 

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

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

⇒ f(x) = (x² – 3x + 2)( x² – 3) 

So, putting x² – 3x + 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, 1, √3 and 2.