<s>[INST] Problem based on Algebraic identities Algebraic identities:
$$(a+b)^2 = a^2 + b^2 + 2ab$$
$$(a-b)^2 = a^2 + b^2 - 2ab$$
Other identities can also come to solve this question?
If $x + 1/x = 5$ and $x^2 + 1/x^3 = 8$, then what would be the value of $x^3 + 1/x^2$?
Possible answers:
a- 215
b- 125
c- 256
d- 525
 [/INST] see ($x^2$+$x^-3$)($x^3$+$x^-2$)


*

*=$x^5$+$x^-5$+2

*NOW x+$\frac{1}{x}$=5. FROM THIS WE GET $x^2$+$x^-2$=23 and $x^3$+$x^-3$=110

*NOW $x^5$+$x^-5$=($x^2$+$x^-2$)($x^3$+$x^-3$)$-$(x+$x^-1$)=23.110$-$5=2525

*SO($x^2$+$x^-3$)($x^3+$$x^-2$)=2525+2=2527

*so $x^3$+$x^-2$=$\frac{2527}{8}${as $x^2$+$x^-3$=8}

</s>