Question

If ‘α’ and ‘β’ are the zeroes of the quadratic polynomial f(x) = 3x2 -7x- 6
find a polynomial whose zeroes are i) 1/ and 1/β

Answer 1

Given equation :- 3x² - 7x + 1

let's factorise it :0 ... by middle term Splitting.

3x² - 7x + 1 = 0

3x² - 3x - x + 1 = 0

3x ( x - 1 ) - 1 ( x - 1 ) = 0

( x - 1 ) ( 3x - 1 ) = 0

° ( x - 1 ) = 0

x = 1

° ( 3x - 1 ) = 0

x = 1/3

• The new quadratic equation have the zeros as

=> alpha² / b and b² /alpha

=> = 1/1/3 = 3

So, the Zeros of new equation are 3 and 1/9

♯ Sum of Zeros :-

3 + 1/9

= ( 27 + 1 )/9

= 28/9

♯ Product of Zeros :-

3 × ( 1/9 )

= 1/3

• To form new quadratic equation we have formula as :-

x² - ( Sum of Zeros )x+ ( Product of Zeros)

x² -28/9 x = 1/3 =0

9x² - 28x + 3 = 0 ( is the required equation )

Answer 2

Step-by-step explanation:

Given -

p(x) = 3x² - 7x - 6

To Find -

A polynomial whose zeroes is 1/α and 1/β

Now,

3x² - 7x - 6

» 3x² + 2x - 9x - 6

» x(3x + 2) - 3(3x + 2)

» (x - 3)(3x + 2)

Zeroes are -

x - 3 = 0 and 3x + 2 = 0

• x = 3 and x = -2/3

Now,

Let α = 3 and β = -2/3

Then,

1/α = 1/3

and

1/β = 1 ÷ -2/3 = -3/2

Sum of zeroes :-

1/3 + (-3/2)

» 1/3 - 3/2

» 2 - 9/6

• » -7/6

And

Product of zeroes :-

1/3 × -3/2

• » -1/2

For a Quadratic polynomial :-

x² - (sum of zeroes)x + product of zeroes

» x² - (-7/6)x + (-1/2)

• » x² + 7/6x - 1/2

Hence,

The polynomial is x² + 7/6x - 1/2