Find all the zeros of the polynomial 2x^4 - 9x^3 + 5x^2 + 3x -1 if two zeros are 2+√3 and 2-√3

Find all the zeros of the polynomial 2x^4 - 9x^3 + 5x^2 + 3x -1 if two

1.	Find all the zeros of the polynomial 2x^4 - 9x^3 + 5x^2 + 3x -1 if two zeros are 2+√3 and 2-√3
Answer» Given:f(x) = (2x4\xa0– 9x3\xa0+ 5x2\xa0+ 3x – 1)Zeroes = (2 + √3) and (2 – √3)Given the zeroes, we can write the factors = (x – 2 + √3) and (x – 2 – √3){Since, If x = a is zero of a polynomial f(x), we can say that x - a is a factor of f(x)}Multiplying these two factors, we can get another factor which is:((x – 2) + √3)((x – 2) – √3) = (x – 2)2 –\xa0(√3)2⇒x2\xa0+ 4 – 4x – 3 = x2\xa0– 4x + 1So, dividing f(x) with (x2\xa0– 4x + 1)f(x) = (x2\xa0– 4x + 1) (2x2\xa0– x – 1)Solving (2x2\xa0– x – 1), we get the two remaining roots as{tex}x = {-b \\pm \\sqrt{b^2-4ac} \\over 2a}{/tex}where f(x) = ax2\xa0+ bx + c = 0(using Quadratic Formula){tex}\\mathrm{x}=\\frac{-(-1) \\pm \\sqrt{(-1)^{2}-4(2)(-1)}}{2(2)}{/tex}{tex}\\mathrm{x}=\\frac{-1 \\pm 3}{4}{/tex}{tex}\\Rightarrow \\mathrm{x}=1,-\\frac{1}{2}{/tex}Zeros of the polynomial =\xa0{tex}1,-\\frac{1}{2}, 2+\\sqrt{3}, 2-\\sqrt{3}{/tex}