Question

If alpha and beta are the zeroes of the Polynomial x^2+ 4x + 3 find the polynomial whose zeroes are 1+beta ÷ alpha and 1 + alpha ÷ beta

Answer 1

α and β are the zeroes of the polynomial x² + 4x + 3.

So,

Sum of zeroes :

α + β = -b/a

⇒ α + β = -4

Product of zeroes:

αβ = c/a

⇒ αβ = 3

Now,

★Sum of zeroes :

1 + β/α + 1 + α/β

⇒ αβ + β² + αβ + α² / αβ

⇒ α² + β² + 2αβ / αβ

We know that,

α² + β² + 2αβ = ( α + β )²

⇒ ( α + β )² / αβ

Putting the values

⇒ ( - 4 )² / 3

⇒ 16 / 3

★ Product of zeroes:

1 + β/α × 1 + α/β

⇒ 1 + α/β + β/α + αβ/αβ

⇒ 2αβ + α² + β² / αβ

⇒ ( α + β )² / αβ

Putting the values

⇒ ( - 4 )² / 3

⇒ 16 / 3

But required polynomial is:

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

⇒ x² - 16/3 x + 16/3

If k = 3

⇒ 3 ( x² - 16/3 x + 16/3 )

⇒ 3x² - 16x + 16

Answer 2

Solution

Given :-

• equation , x² + 4x + 3 = 0
• α & β are two zeroes of this equation .

Find :-

• polynomial equation ,whose zeroes be 1 + β/α & 1 + α/ β

Explanation

Using Formula

★ Sum of zeroes = -(Coefficient of x)/(Coefficient of x²)

★ Product of zeroes = (Constant part)/(coefficient of x²)

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So,

➥ Sum Of Zeroes = -(4)/1

➥ β + α = -4 ------------(1)

And,

➥ Product of zeroes = 3

➥ β . α = 3 -----------(2)

Squaring both side of equ(1)

➥ ( β + α)² = (-4)²

➥ β² + α² + 2 β α = 16

Again, keep Value by equ(2)

➥ β² + α² = 16 - 2 * 3

➥ β² + α² = 16 - 6

➥ β² + α² = 10 ----------(3)

Now, Calculate whose equation ,who have be 1 + β/α & 1 + α/ β

➥ Sum of Zeroes = 1 + β/α + 1 + α/ β

➥ Sum of Zeroes = (2βα + β² + α²)/βα

Keep Value by equ(1) & equ(2)

➥ Sum of Zeroes = (2*3 + 10)/3

➥ Sum of Zeroes = (6+10)/3

➥ Sum of Zeroes = 16/3

Now, again,

➥ Product of zeroes = ( 1 + β/α )*( 1 + α/ β)

➥ Product of zeroes = [(β + α)/α] * [ β + α)/β]

➥ Product of zeroes = (β + α)²/αβ

Keep Value by equ(1) & equ(2)

➥ Product of zeroes = (-4)²/3

➥ Product of zeroes = 16/3

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Formula Of Equation

★ x² - x(Sum of zeroes) + (product of zeroes ) = 0

Keep Values,

➥ x² - x(16/3) + 16/3 = 0

➥ 3x² - 16x + 16 = 0

Hence,Required Equation

• 3x² - 16x + 16 = 0

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