Question

Find all the zeroes of polynomial (2x⁴ – 11x³ + 7x² + 13x – 7), it being given that two of its zeroes are (3 + √2) and (3 – √2).

Answer 1

The given polynomial is f(x) = 2x⁴ – 11x³ + 7x² + 13x – 7. 

Since (3 + √2) and (3 – √2) are the zeroes of f(x) it follows that each one of (x + 3 + √2) and (x + 3 – √2) is a factor of f(x). 

Consequently, [(x – ( 3 + √2)] [(x – (3 – √2)] = [(x – 3) - √2 ] [(x – 3) + √2 ] = [(x – 3)² – 2 ] = x² – 6x + 7, which is a factor of f(x). 

On dividing f(x) by (x² – 6x + 7), we get: