Question

If p = 5 −√21/ 2 , prove that ( p3 + 1/ p3 ) – 5 ( p2 + 1/ p 2 ) + ( p + 1/p ) = 0.

Answer 1

Step-by-step explanation:  

Given If p = 5 −√21/ 2 prove that ( p3 + 1/ p3 ) – 5 ( p2 + 1/ p 2 ) + ( p + 1/p ) = 0.

• Let p = 5 - √21 / 2
• Now 1/p = 2 / 5 - √21
• Rationalising the denominator we get
• 1 / p = 2 / 5 - √21 x 5 + √21 / 5 + √21
•        = 2 (5 + √21) / 25 – 21
•       = 2(5 + √21) / 4
• Or 1/p = 5 + √21 / 2
• Now 1 / p + p = 5 + √21 / 2 + 5 - √21 / 2
•                      = 10 / 2
• So 1/p + p = 5
• Now (p + 1/p)^2 = p^2 + 1/p^2 + 2 p x 1/p
•           (5)^2 = p^2 + 1/p^2 + 2
•  So p^2 + 1/p^2 = 23
• Now p^3 + 1/p^3 = (p + 1/p)(p^2 +1/p^2 – p x 1/p) (using (a + b)^3 )
•                            = 5 (23 – 1)
•                           = 5 x 22
•                           = 110
• Now substituting the values we get
• (p^3 + 1/p^3) – 5(p^2 + 1/p^2) + (p + 1/p)
•        110 – 5(23) + (5)
•           110 – 115 + 5
•                = 0

Reference link wil be

https://brainly.in/question/2081983

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