Question

Obtain all zeroes of
$x {}^{4} + 3x {}^{3} - 2x {}^{2} - 20x - 15.$
If two zeroes are
$\sqrt{5} \: \: and \: \: - \sqrt{5} .$

Answer 1

hope this helps you
regards nishita

Answer 2

Given p(x) = x^4 + 4x^3 - 2x^2 - 20x - 15.

Given that $\sqrt{5}, -\sqrt{5}$ are the zeroes.

Then,

$(x + \sqrt{5})(x - \sqrt{5})$ is also a factor.

$= > x^2 - (\sqrt{5})^2$

$=> x^2 - 5$

------------------------------------------------------------------------------------------------------------

Now,

Divide the given polynomial by x^2 - 5.

x^2 - 5) x^4 + 4x^3 - 2x^2 - 20x - 15  ( x^2 + 4x + 3

            x^4             - 5x^2

           --------------------------------------------

                      4x^3  +  3x^2 - 20x - 15

                      4x^3               - 20x

          ----------------------------------------------

                                    3x^2          -  15

                                    3x^2           -   15

         ------------------------------------------------

                                                0

         ---------------------------------------------------

Now,

We factorize x^2 + 4x + 3

= > x^2 + x + 3x + 3

= > x(x + 1) + 3(x + 1)

= > (x + 1)(x + 3)

= > (x + 1)(x + 3) = 0

= > x = -1,-3 is a zero of p(x).

Therefore, the zeroes of p(x) are:

$=> \boxed{\sqrt{5}, -\sqrt{5}, -1,-3}$

Hope this helps!