Question

Obtain all the zeroes of the polynomial 3x^4-8x^3-12x^2+24x+9;If two of its
Zeroes are √3and -√3
​

Answer 1

Answer :-

√3 , - √3 , - 1/3 , 3

Solution :-

3x^4 - 8x³ - 12x² + 24x + 9

Given :-

√3 and - √3 are zeroes

Finding polynomial of zeroes √3, -√3

Polynomial = ( x - α )( x - β )

= ( x - √3 ){ x - ( - √3 ) }

= ( x - √3 )( x + √3 )

= x² - ( √3 )²

= x² - 3

To find other zeroes divide the given polynomial by ( x² - 3 )

Refer to attachment for division

So, we got the Quotient as 3x² - 8x - 3

Factorising the quotient to find other zeroes

3x² - 8x - 3

= 3x² - 9x + x - 3

= 3x(x - 3) + 1(x - 3)

= (3x + 1)(x - 3)

Zeroes :-

⇒ 3x + 1 = 0 or x - 3 = 0

⇒ 3x = - 1 or x = 3

⇒ x = - 1/3 or x = 3

Therefore √3 , - √3 , - 1/3 and 3 are all the zeroes.