Question

Find all the zeroes of the polynomial 2x⁴ - 9x³ + 5x² + 3x - 1 , if tow of its zeroes are (2 + √3) & (2 - √3) .

Answer with complete steps.

Points : 30 ☺

Answer 1

MULTIPLY THE FACTORS $x-(2+ \sqrt{3} ) AND x-(2-\sqrt{3} )$
WE GET [($(x-2) ^{2}$)-3]
⇒[tex] x^{2} -4x+1 [/tex]
SINCE THIS EQUATION IS ALSO THE FACTOR OF THE GIVEN EQUATION, FOLLOW THE PROCEDURE IN THE PICTURE AND THEN COME BACK AGAIN,
NOW, FACTORIZE $2 x^{2} -x-1$
we get ⇒2x(x-1)+1(x-1)
⇒(x-1)(2x+1)
THEREFORE OTHER FACTORS ARE
x=1 and x=-1/2.
THEREFORE ALL THE ZEROES OF THE GIVEN POLYNOMIAL ARE -1/2,1,$2+ \sqrt{3}$ AND $2- \sqrt{3}$

Answer 2

if two zeros ( 2 +√3) and ( 2 -√3) are given of polynomial it means
{ x -(2 -√3)} and { x -( 2+√3) } are the factors of given polynomial ,

hence,
{ x -(2 -√3)}{x -(2+√3)} is a factor of given polynomial .

{ x² -(2+√3)x -(2-√3)x +(2-√3)(2+√3)} is a factor of given polynomial .

{ x²-(4)x + 1} is a factor of given polynomial .

hence, x²-4x +1 is divisible by given polynomial .

now,

x² -4x +1 ) 2x⁴ -9x³+ 5x² +3x -1( 2x²-x -1

2x⁴ -8x³ +2x²
===============
-x³ +3x² +3x
-x³ +4x² -x
===========
-x² + 4x -1
-x² +4x -1
===========
0000

so, 2x² -x -1 is a factor in which two unknown roots present
now,

2x² -x -1 =0
2x² -2x +x -1 =0
2x( x -1)+ ( x -1) = 0
(2x +1)( x -1)=0
x = -1/2 and 1

so, -1/2 and 1 are two unknown roots of given polynomial .