Given that the a is a root of the equation x2-x-3=0. Evaluate the value of a3+1/a5-a4-a3+a2
Answers
Given : ‘a’ is a root of the equation x² - x - 3 = 0
To find : (a³ + 1) / (a⁵ - a⁴ - a³ + a²)
Solution:
x² - x - 3 = 0
a is a root
=> a² - a - 3 = 0
=> a² - a = 3
=> a(a - 1) = 3
(a³ + 1)/(a⁵ - a⁴ - a³ + a²)
= (a³ + 1) /a²(a³ - a² - a + 1)
= (a³ + 1) /a²(a²(a - 1) -1(a - 1))
using x³ + y³ = (x + y)(x²- xy + y² )
= (a + 1) (a² - a + 1) / a² (a² - 1)(a - 1)
= (a + 1) (a² - a + 1) / a² (a + 1)(a - 1)(a - 1)
= (a² - a + 1) / a²(a - 1)(a - 1)
= (a² - a + 1) / (a(a - 1))²
= (a² - a + 1) / (a² - a)²
using a² - a = 3
= ( 3 + 1)/3²
= 4/9
(a³ + 1)/(a⁵ - a⁴ - a³ + a²) = 4/9
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