Find all zeroes of the polynomial $(2x^4 - 9x^3 + 5x^2 + 3x - 1)$

Find all zeroes of the polynomial $(2x^4 - 9x^3 + 5x^2 + 3x - 1)$ if two of its zeroes are $(2 + \sqrt{3})$ and $(2 - \sqrt{3})$.

Answer

p(x) = 2x⁴ - 9x³ + 5x² + 3x -1

$2 + \sqrt{3}$ and $2 – \sqrt{3}$ are zeroes of p(x) 

p(x) = $(x - 2 - \sqrt{3}) (x - 2 + \sqrt{3}) \times g(x)$

= (x² - 4x + 1) g(x)

(2x⁴ - 9x³ + 5x² + 3x - 1) ÷ (x² - 4x + 1) = 2x² - x - 1 

g(x) = 2x² - x - 1 

= (2x + 1)(x -1)

Therefore other zeroes are x = $-\frac{1}{2}$ and x = 1

Therefore all zeroes are $(2 + \sqrt{3}) and (2 – \sqrt{3}). -\frac{1}{2} and 1$