Question

If α,β,γ be zeros of polynomial 6x3 + 3x2− 5x + 1, then find the value of α−1+β−1+γ−1.​

Answer 1

We have,

p ( x ) = 6x³ + 3x² - 5x + 1

• a = 6

• b = 3

• c = -5

• d = 1

Therefore, α , β and γ are the zeroes of the polynomial.

★ Sum of zeroes:

α + β + γ = -b/a

⇒ α + β + γ = -3/6

⇒ α + β + γ = -1/2 .....( 1 )

★ Sum of the Product of zeroes :

αβ + βγ + αγ = c/a

⇒ αβ + αγ + βγ = -5/6....( 2 )

★ Product of the zeroes :

αβγ = -d/a

⇒ αβγ = -1/6 .....( 3 )

Now,

α−¹ + β−¹ + γ−¹ [ Given ]

⇒ 1/α + + 1/β + 1/γ

⇒ αβ + αγ + βγ / αβγ

⇒ -5/6 / -1/6 [ 2 & 3 ]

⇒ -5/6 × 6/-1

⇒ -5/-1

⇒ 5

Answer 2

$\huge\mathfrak{Answer:}$

Given:

• We have been given that α, β, γ are the zeros of polynomial 6x³ + 3x² - 5x + 1

To Find:

• We need to find the value of α − 1 + β − 1 + γ − 1.

Solution:

The given polynomial is: 6x³ + 3x² - 5x + 1.

α,β,γ are zeros of this polynomial.

Sum of zeroes: (α + β + γ)

= -b/a

= -3/6

= -1/2

Sum of Product of zeroes (αβ + βγ + γα)

= c/a

= -5/6

Product of zeroes (αβɣ)

= -d/a

= -1/6

Now, we need to find the value of

(α⁻¹ + β⁻¹ + ɣ⁻¹), we have

α⁻¹ + β⁻¹ + ɣ⁻¹  

= 1/α + 1/β + 1/ɣ

= (αβ + βɣ + αɣ) / αβɣ

= -5/6 / -1/6

= 5

Hence, the value of α⁻¹ + β⁻¹ + ɣ⁻¹  is 5.