Question

If a, ß are the roots of the equation
x² + 3x + 2 = 0). Then alpha power 5 plus beta power 5 is with neat explanation​

Answer 1

Answer:

        α⁵ + β⁵ = -33

Step-by-step explanation:

Let sₙ = αⁿ + βⁿ.

Then

• s₀ = α⁰ + β⁰ = 1 + 1 = 2
• s₁ = α + β = -3

Notice that the value of -3 for s₁ comes from the fact that

   x² + 3x + 2  =  (x - α)(x - β)  =  x² - (α + β)x + αβ.

Since α is a root of x² + 3x + 2 = 0,

• α² + 3α + 2 = 0   ⇒   α² = -3α - 2  ⇒  αⁿ⁺² = -3αⁿ⁺¹ - 2αⁿ.

Similarly

• βⁿ⁺² = -3βⁿ⁺¹ - 2βⁿ.

Adding these last two equations gives

• sₙ₊₂ = -3sₙ₊₁ - 2sₙ

This enables us to calculate s₂ from our knowledge of s₀ and s₁, and so on for s₃, s₄ and s₅.  Doing this...

• s₂ = -3s₁ - 2s₀  =  -3×-3 - 2×2  =  9 - 4  =  5
• s₃ = -3s₂ - 2s₁  =  -3×5 - 2×-3  =  -15 + 6  =  -9
• s₄ = -3s₃ - 2s₂  =  -3×-9 - 2×5  =  27 - 10  =  17
• s₅ = -3s₄ - 2s₃  =  -3×17 - 2×-9  =  -51 + 18  =  -33

So s₅ = α⁵ + β⁵ = -33.