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abdul9838
@abdul9838
June 2021
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Find all zeros the polynomial of 3x^4+6x^3-2x^2-10x-5 whose to zeros are √5/√3 and - √5/√3
Answers & Comments
Panzer786
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.
1 votes
Thanks 8
shaswata3002
Answer-------
Hope it helps
0 votes
Thanks 3
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Answers & Comments
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-------
Hope it helps