Question

If alpha and beta are the zeros of quadratic polynomial f x is equal to X square + x minus 2 then find the polynomial whose zeros are 2 alpha + 1 and 2 Beta + 1

Answer 1

Solution by finding zeroes :

The given polynomial is

f (x) = x² + x - 2

      = x² + 2x - x - 2

      = x (x + 2) - 1 (x + 2)

      = (x + 2) (x - 1)

We can conclude that (x + 2) and (x - 1) are the factors f (x). Thus, we take (- 2) and 1 as the zeroes of f (x).

     Let, α = - 2 and β = 1

Now, 2α + 1 = 2 (- 2) + 1 = - 4 + 1 = - 3

 and 2β + 1 = 2 ( 1 ) + 1 = 2 + 1 = 3

Thus, the polynomial whose roots are (2α + 1) and (2β + 1) be

g (x) = {x - (2α + 1)} {x - (2β + 1)}

       = {x - (- 3)} (x - 3)

       = (x + 3) (x - 3)

       = x² - 9

Solution by finding the relation between zeroes and coefficients :

The given polynomial is

  f (x) = x² + x - 2

If α and β the zeroes of f (x), then by the relation between zeroes and coefficients, we get

  α + β = - $\frac{1}{1}$ = - 1

  αβ = $\frac{-2}{1}$ = - 2

Hence, the polynomial with the zeroes (2α + 1) and (2β + 1) be

g (x) = x² - {(2α + 1) + (2β + 1)}x + (2α + 1) (2β + 1)

  = x² - {2 (α + β) + 2}x + {4αβ + 2 (α + β) + 1}

  = x² - {2 (- 1) + 2}x + {4 (- 2) + 2 (- 1) + 1}

  = x² - (- 2 + 2)x + (- 8 - 2 + 1)

  = x² - 9

Answer 2

Answer:

Step-by-step explanation:

f (x) = x² + x - 2

      = x² + 2x - x - 2

      = x (x + 2) - 1 (x + 2)

      = (x + 2) (x - 1)

We can conclude that (x + 2) and (x - 1) are the factors f (x). Thus, we take (- 2) and 1 as the zeroes of f (x).

     Let, α = - 2 and β = 1

Now, 2α + 1 = 2 (- 2) + 1 = - 4 + 1 = - 3

 and 2β + 1 = 2 ( 1 ) + 1 = 2 + 1 = 3

Thus, the polynomial whose roots are (2α + 1) and (2β + 1) be

g (x) = {x - (2α + 1)} {x - (2β + 1)}

       = {x - (- 3)} (x - 3)

       = (x + 3) (x - 3)

       = x² - 9

Solution by finding the relation between zeroes and coefficients :

The given polynomial is

  f (x) = x² + x - 2

If α and β the zeroes of f (x), then by the relation between zeroes and coefficients, we get

  α + β = - = - 1

  αβ = = - 2

Hence, the polynomial with the zeroes (2α + 1) and (2β + 1) be

g (x) = x² - {(2α + 1) + (2β + 1)}x + (2α + 1) (2β + 1)

  = x² - {2 (α + β) + 2}x + {4αβ + 2 (α + β) + 1}

  = x² - {2 (- 1) + 2}x + {4 (- 2) + 2 (- 1) + 1}

  = x² - (- 2 + 2)x + (- 8 - 2 + 1)

  = x² -9