Question

cynthia cloned 529 frogs during one year she cloned the same number of frogs each 12 months. Each month she gave 38 of the frogs to her local nature center what is the best estimate for the number of frogs she cloned each month but did not give away

Answer 1

Answer :-

→ The zeroes are α, β, γ = 4, 3, -2 .

Step-by-step explanation :-

Given :-

→ f(x) = x³ - 5x² - 2x + 24

→ The product of its two zeroes is 12 .

To Find :-

→ Zeroes of polynomial [ α, β, γ ] .

Solution :-

→ Let the zeroes of the given cubic polynomial be α , β and γ .

From the given condition we have,

∵ αβ = 12 ..................(1) .

and also we have an identity ,

∵ α + β + γ = - coefficient of x²/coefficient of x³ = -(-5)/1 = 5 ................(2).

∵ αβγ = - constant term/ coefficient of x³ = -24 ......................(3) .

Putting the value of αβ in equation (3), we get

∵ αβγ = -24 .

⇒ 12γ = -24 .

⇒ γ = -24/12 .

∴ γ = -2 ......................(4) .

Putting the value of γ = -2 in equation (2), we get

∵ α + β + γ = 5 .

⇒ α + β + (-2) = 5 .

⇒ α + β = 5 + 2 .

⇒ α + β = 7 ................(5) .

Now,

→ Squaring on both sides, we get

∵ (α + β)² = (7)²

We know the identity [ (α + β)² = (α - β)²+ 4αβ) ]

∴ ( α - β )² + 4 × 12 = 49. [∵ αβ = 12 ]

⇒ (α - β)² + 48 = 49 .

⇒ ( α - β)² = 49 - 48 .

⇒ (α - β)² = 1 .

∴ α - β = 1 ...............(6) .

Now, add in equation (5) and (6), we get

α + β = 7

α - β = 1

+ - +

----------------

⇒ 2α = 8 .

⇒ α = 8/2 .

∴ α = 4 .

Putting α = 4 in equation (5), we get

∵ α - β = 1 .

⇒ 4 - β = 1 .

⇒ - β = 1 - 4 .

⇒ -β = - 3 .

\therefore∴ β = 3 .

∴ The zeroes are α, β, γ = 4, 3, -2 .

Hence, it is solved .