Question

Find all the zeros of 2x 4 -9x 3 +5x 2 +3x-1 if two its zeros are 2+ root 3, 2- root 3 pls answer with explanation

Answer 1

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Answer 2

Value of other two roots are (x-1)(2x+1)

Solution:

Roots of equation are $(x-2+\sqrt{3})(x-2-\sqrt{3})$.

The equation formed is $2 x^{4}-9 x^{3}+5 x^{2}+3 x-1$.  

To find the other two roots we will divide $2 x^{4}-9 x^{3}+5 x^{2}+3 x-1\ by\ x^{2}-4 x+1$

Dividing it by long division method, we get the quotient as  shown in image.

Therefore, the equation formed is $2 x^{2}-x-1$ whose roots are (x-1)(2x+1).  

Multiplying the roots $(x-1)(2 x+1)(x-2+\sqrt{3})(x-2-\sqrt{3})$ will get us $2 x^{4}-9 x^{3}+5 x^{2}+3 x-1.$

Therefore, the value of other two roots are (x-1)(2x+1).