Question

Find all the zeros of of the polynomial 2x^4+7x^3-19x^2-14x+30 if two of its zeroes are root 2 and -root 2

Answer 1

$\textbf{Given:}$

$\text{ \sqrt{2} and -\sqrt{2} are zeros of 2x^4+7x^3-19x^2-14x+30 }$

$\textbf{To find:}$

$\text{zeros of the given polynomial}$

$\textbf{Solution:}$

$\text{Sum of the zeros= \sqrt{2}+(\sqrt{2})=0 }$

$\text{Product of the zeros= (\sqrt{2})(\sqrt{2})=-2 }$

$\text{Corresponding polynomial is}$

$x^2-(0)x-2$

$=x^2-2$

$\text{Now,}$

$2x^4+7x^3-19x^2-14x+30=(x^2-2)(2x^2+px-15)$

$\text{Equating coefficient of x on bothsides, we get}$

$-14=-2\,p$

$\implies\,p=7$

$\implies\text{Other factor is 2x^2+7x-15 }$

$2x^2+7x-15$

$=2x^2+10x-3x-15$

$=2x(x+5)-3(x+5)$

$=(2x-3)(x+5)$

$\implies\,\text{Other zeros are -5 and \frac{3}{2} }$

$\therefore\textbf{All zeros are \bf\sqrt2 , \bf-\sqrt2 , \bf-5 and \bf\frac{3}{2} }$

Find more:

Obtain other zeroes of the polynomial

f(x) = 2x4 + 3x3 - 5x2 - 9x - 3

if two of its zeroes are √3 and - √3.​

https://brainly.in/question/15922200