Question

Find all zeros of polynomial 2x^4+7x^3-19x^2-14x+30 if two of it,s zeros are 2 and 3

Answer 1

Since , two zeroes are √2 and -√2 .

Therefore, (x-√2)(x+√2)=x²-(√2 )²

= x²-2 is a factor of the given polynomial.

Now , we apply the division algorithm to the given polynomial and x²-2.

      2x²+7x-15

     ________________

x²-2)2x⁴+7x³-19x²-14x+30(

*****2x⁴+0-4x²

______________________

********7x³-15x²-14x

********7x³+ 0 -14x

_____________________

***********-15x²+30

********** -15x²+30

_______________________

Remainder ( 0 )

So,

2x⁴+7x³-19x²-14x+30 =(x²-2)(2x²+7x-15)

Now,

2x²+7x-15

Splitting the middle term, we get

= 2x²+10x-3x-15

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

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

So, it's zeroes are x=-5 and x =3/2

Therefore,

The other zeroes of the given polynomial are -5 and 3/2