Question

Obtain all the zeroes of the polynomial f(x) =x4-7x3+10x2-14x-2,if two zeros are √2 and-√2

Answer 1

Hi ,

There is an error in the question .

It may be like this.

It is given that ,

f( x ) = x⁴ - 7x³ + 10x² + 14x - 24

And

√2 , - √2 are two zeroes of f( x ).

( x - √2 ), (x + √2 ) are factors of f(x ).

( x - √2 )( x + √2 ) = x² - ( √2 )²

= x² - 2 is a factor of f( x ).

x²- 2) x⁴ - 7x³ + 10x² +14x - 24(x²-7x+12
*******x⁴ + 0 - 2x²
____________________
***********-7x³ + 12x² +14x
***†*******-7x³ + 0 + 14x
____________________
*****************12x² -24
*****************12x² - 24
____________________
***************†*** 0

Therefore ,

f(x ) = (x² - 2 ) ( x² - 7x + 12 )

= ( x² - 2 ) [ x² - 4x - 3x + 12 ]

= ( x-√2 )( x + √2 ) [ x(x-4)-3(x-4)]

= ( x - √2 )(x + √2 )( x - 4 )( x - 3 )

Required two zeroes are ,

x - 4 = 0 or x - 3 = 0

x = 4 , x = 3

√2 , - √2 , 3 , 4 are zeroes of f( x ).

I hope this helps you.

: )