Cheat Codes: x + 1/x, x – 1/x & Their Powers
November 13, 2023 2025-11-13 17:01Cheat Codes: x + 1/x, x – 1/x & Their Powers
x ± \( \dfrac{1}{x} \) & Powers — One-Page Cheat Code
Master \(x\pm\frac{1}{x}\), \(ax\pm\frac{b}{x}\), and the trig link \(x+\frac{1}{x}=2\cos\theta\). Use a single-variable recurrence to jump to any power fast.
1) Big Idea
Define \(T_n=x^n+\dfrac{1}{x^n}\). If \(x+\dfrac{1}{x}=k\), then
\[
T_0=2,\quad T_1=k,\quad \boxed{T_n=k\,T_{n-1}-T_{n-2}}\ (n\ge2).
\]
So every \(T_n\) is a polynomial in \(k\) — no need to find \(x\).
Cheat: Memorise \(T_n=kT_{n-1}-T_{n-2}\). That’s 90% of the game.
2) When \(x+\dfrac{1}{x}=k\)
Must-know powers (up to 7th)
\[
x^2+\frac{1}{x^2}=k^2-2,\quad
x^3+\frac{1}{x^3}=k^3-3k,\quad
x^4+\frac{1}{x^4}=k^4-4k^2+2,
\]
\[
\boxed{x^5+\frac{1}{x^5}=k^5-5k^3+5k},
\]
\[
\boxed{x^7+\frac{1}{x^7}=k^7-7k^5+14k^3-7k}.
\]
Build from lower powers (handy in options)
\[
\boxed{\,x^5+\frac{1}{x^5}=\Big(x^2+\frac{1}{x^2}\Big)\Big(x^3+\frac{1}{x^3}\Big)-\Big(x+\frac{1}{x}\Big)\,},
\]
\[
\boxed{\,x^7+\frac{1}{x^7}=\Big(x^3+\frac{1}{x^3}\Big)\Big(x^4+\frac{1}{x^4}\Big)-\Big(x+\frac{1}{x}\Big)\,}.
\]
These let you avoid computing huge polynomials when the paper already gives you \(x\pm \tfrac{1}{x}\), \(x^2\pm \tfrac{1}{x^2}\), \(x^3\pm \tfrac{1}{x^3}\), etc.
Connect to minus form
\[
(x+\tfrac{1}{x})^2-(x-\tfrac{1}{x})^2=4
\Rightarrow (x-\tfrac{1}{x})^2=k^2-4
\Rightarrow \boxed{x-\tfrac{1}{x}=\pm\sqrt{k^2-4}}.
\]
Choose sign using any given info about \(x\) (e.g., positivity, quadrant in trig form).
3) When \(x-\dfrac{1}{x}=k\)
Key identities (up to 7th)
\[
x^2+\frac{1}{x^2}=k^2+2,\quad
x^3-\frac{1}{x^3}=k^3+3k,\quad
x^4+\frac{1}{x^4}=k^4+4k^2+2,
\]
\[
\boxed{x^5-\frac{1}{x^5}=k^5+5k^3+5k},
\]
\[
\boxed{x^7-\frac{1}{x^7}=k^7+7k^5+14k^3+7k}.
\]
Back to plus form
\[
(x+\tfrac{1}{x})^2=x^2+\tfrac{1}{x^2}+2=(k^2+2)+2=k^2+4
\Rightarrow \boxed{x+\tfrac{1}{x}=\pm\sqrt{k^2+4}}.
\]
4) General: \(ax\pm\dfrac{b}{x}=k\) (scale to standard)
Use the scaling \(y=\sqrt{\tfrac{a}{b}}\,x\Rightarrow ax\pm \dfrac{b}{x}=\sqrt{ab}\!\left(y\pm \dfrac{1}{y}\right)\).
If \(ax+\dfrac{b}{x}=k\), then
\[
\boxed{y+\frac{1}{y}=\frac{k}{\sqrt{ab}}}\quad\text{with }y=\sqrt{\tfrac{a}{b}}\,x.
\]
If \(ax-\dfrac{b}{x}=k\), then
\[
\boxed{y-\frac{1}{y}=\frac{k}{\sqrt{ab}}}.
\]
Apply Sections 2–3 with \(k\to \dfrac{k}{\sqrt{ab}}\), then convert back if the question demands \(x\).
5) Trig link (why polynomials appear)
If \(x=e^{i\theta}\) then \(\dfrac{1}{x}=e^{-i\theta}\) and
\[
x+\frac{1}{x}=2\cos\theta,\qquad
x^n+\frac{1}{x^n}=2\cos(n\theta).
\]
When \(|k|\le2\), write \(k=2\cos\theta\) to jump powers via \(2\cos(n\theta)\) (Chebyshev viewpoint).
6) Cheat Codes at a Glance
| Given | Key outcomes (single-variable) |
|---|---|
| \(x+\dfrac{1}{x}=k\) |
\(x^2+\dfrac{1}{x^2}=k^2-2\),; \(x^3+\dfrac{1}{x^3}=k^3-3k\),; \(x^4+\dfrac{1}{x^4}=k^4-4k^2+2\),; \(x^5+\dfrac{1}{x^5}=k^5-5k^3+5k\),; \(x^7+\dfrac{1}{x^7}=k^7-7k^5+14k^3-7k\). Also: \(x^5+\dfrac{1}{x^5}=(x^2+\dfrac{1}{x^2})(x^3+\dfrac{1}{x^3})-(x+\dfrac{1}{x})\); \(x^7+\dfrac{1}{x^7}=(x^3+\dfrac{1}{x^3})(x^4+\dfrac{1}{x^4})-(x+\dfrac{1}{x})\). Recurrence: \(T_n=kT_{n-1}-T_{n-2}\). |
| \(x-\dfrac{1}{x}=k\) |
\(x^2+\dfrac{1}{x^2}=k^2+2\),; \(x^3-\dfrac{1}{x^3}=k^3+3k\),; \(x^4+\dfrac{1}{x^4}=k^4+4k^2+2\),; \(x^5-\dfrac{1}{x^5}=k^5+5k^3+5k\),; \(x^7-\dfrac{1}{x^7}=k^7+7k^5+14k^3+7k\). |
| \(ax+\dfrac{b}{x}=k\) | Let \(y=\sqrt{\tfrac{a}{b}}x\Rightarrow y+\dfrac{1}{y}=\dfrac{k}{\sqrt{ab}}\). Apply the first row to \(y\). |
| \(ax-\dfrac{b}{x}=k\) | Let \(y=\sqrt{\tfrac{a}{b}}x\Rightarrow y-\dfrac{1}{y}=\dfrac{k}{\sqrt{ab}}\). Apply the second row to \(y\). |
| \(x+\dfrac{1}{x}=2\cos\theta\) | \(x^n+\dfrac{1}{x^n}=2\cos(n\theta)\) (clean power jumps). |
7) Quick Drill (CAT-style)
Q1. If \(x+\dfrac{1}{x}=5\), find \(x^5+\dfrac{1}{x^5}\).
\[ x^5+\frac{1}{x^5}=k^5-5k^3+5k=3125-625+25=\boxed{2525}. \]Q2. If \(x-\dfrac{1}{x}=3\), find \(x^5-\dfrac{1}{x^5}\).
\[ x^5-\frac{1}{x^5}=k^5+5k^3+5k=243+135+15=\boxed{393}. \]Q3. If \(x+\dfrac{1}{x}=3\), compute \(x^7+\dfrac{1}{x^7}\).
\[ x^7+\frac{1}{x^7}=k^7-7k^5+14k^3-7k =2187-7(243)+14(27)-21=\boxed{843}. \]Q4. If \(x-\dfrac{1}{x}=2\), compute \(x^7-\dfrac{1}{x^7}\).
\[ x^7-\frac{1}{x^7}=k^7+7k^5+14k^3+7k =128+7(32)+14(8)+14=\boxed{478}. \]© AzuCATion — “Think CAT, Think AzuCATion.”
