Question

6. Find all the zeroes of 2x^4 - 3x^3 - 5x^2 + 9x - 3,
if its two zeroes are √3 , - √3.

Class 10

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Answer 1

Given f(x) = 2x^4 - 3x^3 - 5x^2 + 9x - 3.

Given $(x + \sqrt{3} (x - \sqrt{3} )$ is a factor.

We know that (a + b)(a - b) = a^2 - b^2

= > (x^2 - 3) is a factor

Divide the given polynomial by (x^2 - 3).



               2x^2 - 3x + 1
             ------------------------------------
x^2 - 3) 2x^4 - 3x^3 - 5x^2 + 9x - 3

            2x^4 -           - 6x^2

            -----------------------------------------

                       -3x^3 +  x^2 + 9x - 3

                       - 3x^3           + 9x

             -----------------------------------------

                                      x^2          - 3

                                      x^2          - 3

                -------------------------------------------

                                                     0


Now,

we should factorize 2x^2 - 3x + 1

= > 2x^2 - 3x + 1

= > 2x^2 - 2x - x + 1

= > 2x(x - 1) - 1(x - 1)

= > (2x - 1)(x - 1)

= > x = 1/2, 1.



Therefore the zeroes of the polynomial are: 

$= \ \textgreater \ \sqrt{3} , - \sqrt{3} , \frac{1}{2} , 1$



Hope this helps!

Answer 2

Let, m , n , o and p be its zeroes.

Given,

Quartic equation = 2x⁴ - 3x³ - 5x² + 9x - 3

Two zeroes are √3 and -√3, where ( √3 = m ) and ( -√3 = n ).

Here,

Coefficient of x⁴ ( a ) = 2

Coefficient of x³ ( b ) = -3

Coefficient of x² ( c ) = -5

Coefficient of x ( d ) = 9

Constant term ( e ) = -3

We know that,

⇒ Sum of zeroes of a quartic equation = -( Coefficient of x³ ) ÷ ( Coefficient of x⁴ )

⇒ m + n + o + p = -b/a

⇒ √3 + ( -√3 ) + o + p = -( -3 )/2

⇒ √3 - √3 + o + p = 3/2

•°• o + p = ( 3/2 ) --------------- ( 1 )

Now,

⇒ Product of zeroes = Constant term / Coefficient of x⁴

⇒ mnop = e/a

⇒ ( √3 ) ( -√3 ) op = -3/2

⇒ -3op = -3/2

⇒ op = -3 / ( 2 × -3 )

•°• op = ( 1/2 ) --------------- ( 2 )

Using identity,

⇒ ( a - b )² = ( a + b )² - 4ab


⇒ ( o - p )² = ( o + p )² - 4op

Substitute the value of ( 1 ) and ( 2 ),

⇒ ( o - p )² = ( 3/2 )² - 4( 1/2 )

⇒ ( o - p )² = ( 9/4 ) - 2

⇒ ( o - p )² = ( 9 - 8 ) / 4

⇒ ( o - p )² = 1/4

⇒ ( o - p ) = √( 1/4 )

•°• ( o - p ) = ( 1/2 ) ------------ ( 3 )

Adding ( 1 ) and ( 3 ),

⇒ o + p + o - p = ( 3/2 ) + ( 1/2 )

⇒ 2o = ( 3 + 1 ) /2

⇒ 2o = 4/2

⇒ 2o = 2

⇒ o = 2÷2

•°• o = 1

Plug the value of o in ( 3 ),

⇒o - p = 1/2

⇒ 1 - p = 1/2

⇒ 1 - ( 1/2 ) = p

⇒ ( 2 - 1 ) / 2 = p

•°• p = 1/2

Hence, two other zeroes are 1 and ( 1/2 ).