Question

if root 2 and minus root 2 are two zeros of the polynomial x to the power 4 plus x to the power cube minus 14 x to the power square - 2 X + 24 then its other two zeros are​

Answer 1

GIVEN :

The roots $\sqrt{2}$ and $-\sqrt{2}$  are two zeroes of the polynomial $x^4+x^3-14 x^2- 2x+24$

TO FIND :

The other two zeroes of the given polynomial.

SOLUTION :

Given that the roots $\sqrt{2}$ and $-\sqrt{2}$  are two zeroes of the polynomial $x^4+x^3-14 x^2- 2x+24$

Since $\sqrt{2}$ and $-\sqrt{2}$  are roots , we can write it as,

$(x-\sqrt{2})(x+\sqrt{2})=0$

$(x^2-\sqrt{2}^2=0$

$x^2-2=0$

Hence $x^2-2$ is the factor of the given polynomial.

                            $x^2+x-12$

                        _______________________

             $x^2-2$ )  $x^4+x^3-14 x^2- 2x+24$

                           $x^4-2x^2$

                         _(-)__(+)________

                                 $x^3-12x^2$

                                 $x^3-2x^2$

                           ___(-)__(+)________

                                     $-12x^2+24$

                                     $-12x^2+24$

                                 __(+)____(-)____  

                                                  0

                                      ___________

Hence the quotient is $x^2+x-12$

∴ $x^2+x-12$ is the factor of the given polynomial. So we can write it as

$x^2+x-12=0$

$(x+4)(x-3)=0$

x+4=0 or x-3=0

∴ x=-4 and  x=3 are the other two zeroes of the given polynomial