Question

A (Alpha square upon beta + beta square upon alpha) + b (Alpha upon beta + beta cube upon Alpha)​

Answer 1

We have to simplify the given term to smaller form:

Now, a (α²/β² + β²/α²) + b (α³/β³ + β³/α³)

= a {(α/β + β/α)² - 2} + b {(α/β + β/α)³ - 3 (α/β + β/α)}

= a (γ² - 2) + b (γ³ - 3γ),

where α/β + β/α = γ ( assumed )

= aγ² - 2a + bγ³ - 3bγ

= bγ³ + aγ² - 3bγ - 2a

This is the required simplification.

Algebraic identities:

• a² + b² = (a + b)² - 2ab
• a³ + b³ = (a + b)³ - 3ab (a + b)
• a² - b² = (a + b) (a - b)
• (a + b)³ = a³ + 3a²b + 3ab² + b³