Calculate x³+1/x³ Given x+1/x=5: No-Solving Method

We want to calculate the value of the expression:

$x^3+\frac{1}{x^3}=\text{?}$

Based on the given equation:

$x+\frac{1}{x}=5$

but without solving it for x,

For this, let's first note that while the given equation deals with terms with first power only,

in the expression we want to calculate - there are terms with third power only,

However to start let's note additionally that the value of the expression:

$x^2+\frac{1}{x^2}=\text{?}$can be calculated more easily, since it involves only terms with second power:

Therefore, we understand that to do this - apparently we need to square the expression on the left side of the given equation,

We'll recall of course the formula for squaring a binomial:

$(a\pm b)^2=a^2\pm2ab+b^2$and we'll square both sides of the given equation, then we'll highlight something worth noting that happens in the given mathematical structure in the expression in question (the mathematical structure where a term and its opposite are added):

$x+\frac{1}{x}=5 \hspace{6pt}\text{/}()^2\\ (x+\frac{1}{x})^2=5^2\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{x\cdot \frac{1}{x}}+ \frac{1}{x^2}=25\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{1}+ \frac{1}{x^2}=25\\$Let's now note that the "mixed" term in the square formula ($2ab$) gives us - from squaring the mathematical structure in question - a free number, meaning - one that doesn't depend on the variable x, since it's a multiplication between an expression and its opposite,

This fact actually allows us to isolate the desired expression from the equation we get and find its value (which doesn't depend on the variable) even without knowing the value(s) of the unknown(s) that solve the equation:

$x^2+2\cdot \textcolor{blue}{1}+ \frac{1}{x^2}=25\\ x^2+2+ \frac{1}{x^2}=25\\ \boxed{x^2+\frac{1}{x^2}=23}$

From here we'll continue and return to our goal- calculating the value of the expression:

$x^3+\frac{1}{x^3}=\text{?}$Let's note that if we multiply both sides of the last equation we got by the expression: $x+\frac{1}{x}$we can get on the left side an expression containing the desired terms (with third power) and additionally the value of this expression we already know since the given equation is:

$\textcolor{blue}{x+\frac{1}{x}=5}$Let's do this then, and afterwards we'll try to isolate the desired expression from the resulting equation:

$$$x^2+\frac{1}{x^2}=23\hspace{6pt}\text{/}\cdot(x+\frac{1}{x})\\ (x+\frac{1}{x})(x^2+\frac{1}{x^2})=(x+\frac{1}{x})\cdot23$

From here we'll continue in two different ways, on the left side, from which we want to get the desired terms (with third power), we'll prefer to expand the parentheses using the expanded distribution law and simplify later, but on the right side, where we don't want to have dependency on the unknown, we'll substitute what we know from the given equation (highlighted in blue):

$(x+\frac{1}{x})(x^2+\frac{1}{x^2})=\textcolor{blue}{(x+\frac{1}{x})}\cdot23\\ \downarrow\\ x^3+\frac{1}{x}+x+\frac{1}{x^3}=\textcolor{blue}{5}\cdot23\\ x^3+\frac{1}{x^3}+x+\frac{1}{x}=115\\$$$Now we'll identify again that the sum of the last two terms in the expression on the left side - we already know (again - from the given equation - highlighted in blue), and therefore we can substitute this information again, isolate the desired expression and get its value - without dependency on the unknown x:

$x^3+\frac{1}{x^3}+\textcolor{blue}{\underline{x+\frac{1}{x}}}=115\\ \downarrow\\ x^3+\frac{1}{x^3}+\textcolor{blue}{\underline{5}}=115\\ \boxed{x^3+\frac{1}{x^3}=110}$

Therefore the correct answer is answer B.