Question

obtain all the other zeroes of the polynomial p (x) = 2x4-6x3+3x2+3x-2 if two of zeroes are 1 / root 2 and -1 / root 2 ?

Answer 1

So, polynomial P(x)= $2x^{4}-6x^{3}+3x^{2}+3x-2$ and it's zeroes are 1/√2 and -1/√2. We need to find all other zeroes. Let x be the all other zeroes.

(x-1/√2)(x+1/√2)

(x²-1/2)=0

(2x²-1)=0

Hence, the factor is 2x²-1. Therefore,

                      $x^{2} -3x+2$

           ___________________________

$2x^{2} -1$     $2x^{4}-6x^{3}+3x^{2}+3x-2$

               $2x^{4}-x^{2}$

            ___________________________

                      $-6x^{3}+4x^{2}+3x-2$

                      $-6x^{3}+3x$

                     ______________________

                                   $4x^{2}-2$

                                   $4x^{2}-2$

                               _________________

                                                       x

Now, $2x^{4}-6x^{3}+3x^{2}+3x-2$ = $(2x^{2}-1)(x^{2}-3x+2)$

On future factorizing $(x^{2}-3x+2)$ we get,

⇒ $x^{2}-2x-x+2=0$

⇒ $x(x-2)-1(x-2)=0$

⇒ $(x-2)(x-1)=0$

∴ x=2 and x=1

2 and 1 are other zeroes of the polynomial.