Question

Find all zeros the polynomial of 3x^4+6x^3-2x^2-10x-5 whose to zeros are √5/√3 and - √5/√3

Answer 1

Heya !!

Given that : ✓5 / 3 and - ✓5 / 3 are two zeroes of the given polynomial.

( X - ✓5/3 ) and ( X + ✓5/3) are also a factor of the given polynomial.

Therefore,

( X - ✓5/3 ) ( X + ✓5/3) = ( X )² - ( ✓5/3)²

=> X² - 5/3

=> 3X² - 5 is a factor of the given polynomial.

G ( X ) = 3X² - 5

And,

P ( X ) = 3X⁴ + 6X³ - 2X² - 10X - 5

On dividing P ( X ) by G ( X ) we get ,

3X²-5 ) 3X⁴+6X³-2X² -10X - 5 (X²+2X +1

**********3X⁴**********-5X²

----------------------------------------------------

*************+6X³ ****+3X² - 10X - 5

**************+6X³ **********-10X

______________________________

***********************+3X² ********-5

************************+3X² *******-5

______________________________

**************************0************0

Remainder = 0

and,

Quotient = X² + 2X + 1

=> X² + X + X + 1

=> X ( X + 1 ) + 1 ( X + 1 )

=> ( X + 1 ) ( X + 1 )

=> ( X + 1 ) = 0

=> X = -1

Hence,

All four zeroes of the given polynomial are ✓5/3 , -✓5/3 , -1 and -1.

Answer 2

Answer-------

Hope it helps