Question

Α β γ are zeroes of cubic polynomial x3 12x2 44x c if α β γ are in ap find the value of c

Answer 1

$\textbf{Answer -}$

The given polynomial is

p (x) = x³ + 12x² + 44x + c

Since, α, β and γ are the roots of p (x),

α + β + γ = - 12 ...(i)

αβ + βγ + γα = 44 ...(ii)

αβγ = - c ...(iii)

Given that, α, β and γ are in AP,

β - α = γ - α

or, 2α = β + γ ...(iv)

Now, (i) × 2 ⇒

2α + 2 (β + γ) = - 24

⇒ (β + γ) + 2 (β + γ) = - 24

⇒ 3 (β + γ) = - 24

⇒ β + γ = - 8 ...(v)

Now, putting β + γ = - 8 in (i), we get

α - 8 = - 12

⇒ α = - 12 + 8

⇒ α = - 4

Since α (= - 4) is a zero of p (x),

p (α) = 0

⇒ p (- 4) = 0

⇒ (- 4)³ + 12 (- 4)² + 44 (- 4) + c = 0

⇒ - 64 + 192 - 176 + c = 0

⇒ c = 48

Therefore, the value of c is 48.

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Answer 2

Answer:

Step-by-step explanation:

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Secondary School Math 13+7 pts

Α β γ are zeroes of cubic polynomial x3 12x2 44x c if α β γ are in ap find the value of c

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\textbf{Answer -}

The given polynomial is

p (x) = x³ + 12x² + 44x + c

Since, α, β and γ are the roots of p (x),

α + β + γ = - 12 ...(i)

αβ + βγ + γα = 44 ...(ii)

αβγ = - c ...(iii)

Given that, α, β and γ are in AP,

β - α = γ - α

or, 2α = β + γ ...(iv)

Now, (i) × 2 ⇒

2α + 2 (β + γ) = - 24

⇒ (β + γ) + 2 (β + γ) = - 24

⇒ 3 (β + γ) = - 24

⇒ β + γ = - 8 ...(v)

Now, putting β + γ = - 8 in (i), we get

α - 8 = - 12

⇒ α = - 12 + 8

⇒ α = - 4

Since α (= - 4) is a zero of p (x),

p (α) = 0

⇒ p (- 4) = 0

⇒ (- 4)³ + 12 (- 4)² + 44 (- 4) + c = 0

⇒ - 64 + 192 - 176 + c = 0

⇒ c = 48

Therefore, the value of c is 48