Question

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Answer 1

Answer:

hope it's helpful to you...

Answer 2

Question :

If a + b = - c, then prove that a³ + b³ + c³ = 3abc

Given :

• a + b = - c

To prove :

• a³ + b³ + c³ = 3abc

Knowledge required :-

Identity to be used :-

• (a + b)³ = a³ + b³ + 3ab(a + b)

Solution :

⠀⠀⠀⇒ a + b = - c

⠀⠀⠀⇒ Cubing both the sides.

⠀⠀⠀⇒ (a + b)³ = (- c)³

⠀⠀⠀⇒ (a)³ + (b)³ + 3ab(a + b) = - c³

⠀⠀⠀⇒ a³ + b³ + 3ab(- c) = - c³

⠀⠀⠀⇒ a³ + b³ + (- 3abc) = - c³

⠀⠀⠀⇒ a³ + b³ - 3abc = - c³

⠀⠀⠀⇒ a³ + b³ + c³ = 3abc

Hence, proved.

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Some useful identities :-

• (a + b)² = a² + b² + 2ab
• (a - b)² = a² + b² - 2ab
• a² - b² = (a + b)(a - b)
• (a - b)³ = a³ - b³ - 3ab (a - b)
• (a + b)³ = a³ + b³ + 3ab (a + b)
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a + b - c)² = a² + b² + c² + 2ab - 2bc - 2ca
• (a - b + c)² = a² + b² + c² - 2ab - 2bc + 2ca
• (- a + b + c)² = a² + b² + c² - 2ab + 2bc + 2ca
• (a - b - c)² = a² + b² + c² - 2ab + 2bc - 2ca
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)
• If a + b + c = 0 then,
• a³ + b³ + c³ = 3abc

Signs are changed on the following bases :-

• (-) (-) = (+)
• (+) (+) = (+)
• (-) (+) = (-)
• (+) (-) = (-)