Question

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Answer 1

See the attachment for the calculation.

The required polynomial is 
=x²+4x+4

Answer 2

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Given,

Quadratic equation = x² - 1.

α and ß are its zeroes.

1st Method :

= x² - 1

= ( x )² - ( 1 )²

= ( x + 1 ) ( x - 1 )

So, zeroes = 1 and -1.

α = 1 and ß = -1.

The general form of a quadratic equation is :

= x² - ( Sum of zeroes ) x + Product of zeroes

= x² - { ( 2α/ß ) + ( 2ß/α ) }x + ( 2α/ß ) ( 2ß/α )

By substituting the values of α and ß,

= x² - { ( 2× 1 / -1 ) + ( 2 × -1 / 1 ) } x + ( 2×1 / -1 ) ( 2 × -1 / 1 )

= x² - { -2 - 2 }x + ( -2 ) ( -2 )

= x² - ( -4 )x + 4

= x² + 4x + 4.

2nd method :

Quadratic equation = x² - 1, α and ß are its zeroes.

Here,

Coefficient of x² ( a ) = 1

Coefficient of x ( b ) = 0

Constant term ( c ) = -1

Sum of zeroes = -b/a

 α + ß = -0/1

  α + ß = 0

Now,

Product of zeroes = c/a

 αß = -1 / 1

 αß = -1.

The general form of a quadratic equation is :

= x² - ( Sum of zeroes ) x + Product of zeroes

= x² - { ( 2α/ß ) + ( 2ß/α ) }x + ( 2α/ß ) ( 2ß/α )

= x² - { ( 2α² + 2ß² ) / αß }x + 4

= x² - { 2 ( α² + ß² ) / -1 }x + 4

= x² - [ 2 { ( α + ß )² - 2αß } / -1 ] x + 4

= x² - [ 2 { 0² - 2 ( -1 ) } / -1 ] x + 4

= x² - [ 2 { 2 } / -1 ] x + 4

= x² - ( -4 ) x + 4

= x² + 4x + 4.

The required quadratic polynomial is x² + 4x + 4.