Answer:

Required equation is 25x^2 - 30x + 4.

Required POLYNOMIAL is 25x^2 - 30x + 4 .

Step-by-step explanation:

MAY be : Given method is not in your book.

ACCORDING to this question :

Zeroes of an equation are ( 3 + √5 ) / 5 and ( 3 - √5 ) / 5 .

We know, every quadratic equation can be expressed in the form of ( x - a )( x - b ) = 0 , where a and b are the zeroes of that equation.

So, if ( 3 + √5 ) / 5 and ( 3 - √5 ) / 5 are zeroes :

= > { x - ( 3 + √5 ) / 5 }{ x - ( 3 - √5 ) / 5 } = 0

= > { ( 5x - 3 - √5 ) / 5 }{ ( 5x - 3 + √5 ) / 5 } = 0

= > ( 5x - 3 - √5 )( 5x - 3 + √5 ) = 0

= > 5x( 5x - 3 + √5 ) - 3( 5x - 3 + √5 ) - √5( 5x - 3 + √5 ) = 0

= > 25x^2 - 15X + 5√5 x - 15x + 9 - 3√5 - 5√5 x + 3√5 - 5 = 0

= > 25x^2 - 30x + 4 = 0

Hence the required equation is 25x^2 - 30x + 4 = 0.

Method 2

Given zeroes : ( 3 - √5 ) / 5 and ( 3 + √5 ) / 5 .

Sum of zeroes :

= > ( 3 - √5 ) / 5 + ( 3 + √5 ) / 5

= > ( 3 - √5 + 3 + √5 ) / 5

= > 6 / 5

Product of zeroes :

= > ( 3 - √5 ) / 5 ( 3 + √5 ) / 5

= > ( 3 - √5 )( 3 + √5 ) / 25

= > ( 9 - 5 ) / 25 { ( a + b )( a - b ) = a^2 - b^2 }

= > 4 / 25

From the properties of quadratic equations :

Quadratic equation : x^2 - ( sum of zeroes )x + product of zeroes = 0

= > x^2 - ( 6 / 5 )x + ( 4 / 25 ) = 0 ... { from above }

= > ( 25x^2 - 30x + 4 ) / 25 = 0

= > 25x^2 - 30x + 4 = 0

Hence the required equation is 25x^2 - 30x + 4 = 0.

And required polynomial is 25x^2 - 30x + 4 .