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

1.	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» (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.

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»

(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.