If x + 1/ x = a, then what is the value of x^3 + x^2 + 1/x^3 + 1/x^2 ?

Question

If \(x+\frac{1}{x}=a\),  then what is the value of x³ + x² + \(\frac{1}{x^3}+\frac{1}{x^2}\) ?

(a) a³ + a² 

(b) a³ + a²  – 5a 

(c) a³ + a²  – 3a – 2 

(d) a³ + a² – 4a – 2

Answer 1

(c) a³ + a²  – 3a – 2

Given, \(x+\frac{1}{x}=a\) 

Now,  x³ + x² + \(\frac{1}{x^3}+\frac{1}{x^2}\) = \(\big(x^3+\frac{1}{x^3}\big)\)+ \(\big(x^2+\frac{1}{x^2}\big)\)

= \(\big(x+\frac{1}{x}\big)^3\) - 3\(\big(x+\frac{1}{x}\big)\) + \(\big(x+\frac{1}{x}\big)^2\) - 2

= a³ – 3a + a² – 2 = a³ + a² – 3a – 2.