Question

Write the quadratic polynomial whose zeros are 1/α and 1/β

Answer 1

Step-by-step explanation:

Given -

• Zeroes of the polynomial is 1/α and 1/β

To Find -

• A quadratic polynomial

Method 1 :-

As we know that :-

• α + β = -b/a

→ 1/α + 1/β = -b/a

→ β + α/αβ = -b/a ....... (i)

And

• αβ = c/a

→ 1/α × 1/β = c/a

→ 1/αβ = c/a ...... (ii)

Now, From (i) and (ii), we get :

a = αβ

b = -(α + β)

c = 1

Now,

As we know that :-

For quadratic polynomial :

• ax² + bx + c

→ αβx² - (α + β)x + 1

→ αβx² - (α + β)x + 1

The quadratic polynomial is αβx² - (α + β)x + 1

Method 2 :-

As we know that :-

For a quadratic polynomial :

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

→ x² - [(α + β)x/αβ] + 1/αβ = 0

→ αβx² - (α + β)x + 1/αβ = 0

→ αβx² - (α + β)x + 1 = 0

Hence,

The quadratic polynomial is αβx² - (α + β)x + 1

Answer 2

$\huge \mathfrak{Answer:-}$

$\large\underline\mathrm{The \: quadratic \: polynomial \: is βαx² \: -(α \: + \: β)x \: + \: 1.}$

$\large\underline\mathrm{Given:-}$

• zeros are 1/α and 1/β

$\large\underline\mathrm{To \: find}$

• A quadratic polynomial.

$\large\underline\mathrm{Solution}$

• 1/α + 1/β = -b/a

$\implies$ 1/α + 1/β = -b/a

$\implies$ β + α/βα = -b/a. ...(1)

$\large\underline\mathrm{and}$

• βα = c/a

$\implies$ 1/α × 1/β = c/a

$\implies$ 1/βα = c/a. ....(2)

$\large\underline\mathrm{Now,}$

$\large\underline\mathrm{Eq. \: (1) \: \: and \: Eq. \: (2), \: we \: get;}$

$\implies$ a = βα

$\implies$ b = -(α + β)

$\implies$ c = 1

$\large\underline\mathrm{Thus,}$

$\large\underline\mathrm{using \: the \: formula \: in \: quadratic \: polynomial \: ;}$

$\implies$ ax² + bx - c

$\implies$ βαx² -(α + β)x + 1

$\large\underline\mathrm{The \: \: quadratic \: polynomial;}$

$\implies$ αβx² - (α + β)x + 1

$\large\underline\mathrm{Thus,}$

$\large\underline\mathrm{for \: the \: quadratic \: polynomial.}$

$\implies$ x² - (sum of zeroes)x + (products of zeroes)

$\implies$ x² - [(α + β)x/αβ] + 1/βα = 0

$\implies$ βαx² -(α + β)x + 1/βα

$\implies$ βαx² -(α + β)x + 1 = 0

$\large\underline\mathrm{hence,}$

$\large\underline\mathrm{The \: quadratic \: polynomial \: is βαx² \: -(α \: + \: β)x \: + \: 1.}$

$\large\underline\mathrm{Hope \: it \: helps \: you \: plz \: mark \: me \: brainlist}$