Question

Find all the zeros of 2x to the power 4 -13x cube +19xsquare+7x-3, if you know that two of its zeros are 2+root3 &2-root3.

Answer 1

EXPLANATION.

• GIVEN

Zeroes of the polynomial =

=> 2x⁴ - 13x³ + 19x² + 7x - 3 it's two zeroes are

=> 2 + √3 and 2 - √3

zeroes of the polynomial

=> x = 2 + √3 and x = 2 - √3

=> x - ( 2 + √3 ) and x - (2 - √3 )

=> x - 2 - √3 and x - 2 + √3

=> products of the zeroes

=> ( x - 2 )² - (√3)²

=> x² + 4 - 4x - 3

=> x² - 4x + 1

on dividing the polynomial by = x² - 4x + 1

we get,

=> 2x² - 5x - 3

=> Factories into middle term splits.

we get,

=> 2x² - 5x - 3 = 0

=> 2x² - 6x + x - 3 = 0

=> 2x ( x - 3 ) + 1 ( x - 3 ) = 0

=> ( 2x + 1 ) ( x - 3 ) = 0

=> x = -1/2 and x = 3

Therefore,

All zeroes are =

=> ( 2 + √3 ) , ( 2 - √3 ), -1/2 , 3