Given that x root 5 is a factor of the cubic polynomial x3 3 root5x2 + - Maths - Polynomials

Question

Given that x – root 5 is a factor of the cubic polynomial
x3 – 3 root5x2 + 13x – 3 root5 ,
find all the zeroes of the polynomial

Shridhar Kaushik · Accepted Answer

Given polynomial is $p (x) = x^{3} - 3 \sqrt{5} x^{2} + 13 x - 3 \sqrt{5}$ Also $(x - \sqrt{5})$ is a factor if p(x)

$∴ p (x) = (x - \sqrt{5}) (x^{2} - 2 \sqrt{5} x + 3)$ For finding the zeroes of the polynomial p(x), we have $\Rightarrow x - \sqrt{5} = 0 o r x^{2} - 2 \sqrt{5} x + 3 = 0 \Rightarrow x = \sqrt{5} or x = \frac{2 \sqrt{5} \pm \sqrt{20 - 12}}{2 \times 1} \Rightarrow x = \sqrt{5} or x = \frac{2 \sqrt{5} \pm 2 \sqrt{2}}{2} \Rightarrow x = \sqrt{5} or x = \sqrt{5} + \sqrt{2} or x = \sqrt{5} - \sqrt{2}$

Given that x – root 5 is a factor of the cubic polynomial x3 – 3 root5x2 + 13x – 3 root5 , find all the zeroes of the polynomial.

Given that x – root 5 is a factor of the cubic polynomial x3 – 3 root5x2 + 13x – 3 root5 ,

find all the zeroes of the polynomial.