🎯 Symmetry Substitutions — u=x+1/x, t=x², Cyclic Sums

Essential technique for solving complex algebraic problems with symmetry.

🎯 Recognition Cues

  • “Simplify” — Expressions with $x$ and $\frac{1}{x}$
  • “Find the value of” — Complex expressions that can be simplified
  • $x + \frac{1}{x}$ — Symmetric expressions
  • $x^2 + \frac{1}{x^2}$ — Powers of symmetric expressions
  • Cyclic expressions — $x + y + z$ where $x, y, z$ are related

📚 Template Solutions

Common Substitutions

PatternSubstitutionPurposeExample
$x + \frac{1}{x}$$u = x + \frac{1}{x}$Simplify expressions$x^2 + \frac{1}{x^2} = u^2 - 2$
$x^2$$t = x^2$Reduce degree$x^4 + x^2 + 1 = t^2 + t + 1$
$x + y + z$$s = x + y + z$Cyclic sums$x^2 + y^2 + z^2 = s^2 - 2(xy + yz + zx)$
$\sqrt{x}$$u = \sqrt{x}$Eliminate radicals$x + \sqrt{x} = u^2 + u$

Symmetry Identities

ExpressionIdentityExample
$x^2 + \frac{1}{x^2}$$(x + \frac{1}{x})^2 - 2$$x^2 + \frac{1}{x^2} = u^2 - 2$ where $u = x + \frac{1}{x}$
$x^3 + \frac{1}{x^3}$$(x + \frac{1}{x})^3 - 3(x + \frac{1}{x})$$x^3 + \frac{1}{x^3} = u^3 - 3u$ where $u = x + \frac{1}{x}$
$x^4 + \frac{1}{x^4}$$(x^2 + \frac{1}{x^2})^2 - 2$$x^4 + \frac{1}{x^4} = (u^2 - 2)^2 - 2$ where $u = x + \frac{1}{x}$

🎯 Worked Examples

Example 1: Basic Symmetry

Problem: If $x + \frac{1}{x} = 3$, find $x^2 + \frac{1}{x^2}$.

Solution:

  1. Let $u = x + \frac{1}{x}$: $u = 3$
  2. Use identity: $x^2 + \frac{1}{x^2} = u^2 - 2$
  3. Substitute: $x^2 + \frac{1}{x^2} = 3^2 - 2 = 9 - 2 = 7$
  4. Answer: $7$

Example 2: Higher Powers

Problem: If $x + \frac{1}{x} = 2$, find $x^3 + \frac{1}{x^3}$.

Solution:

  1. Let $u = x + \frac{1}{x}$: $u = 2$
  2. Use identity: $x^3 + \frac{1}{x^3} = u^3 - 3u$
  3. Substitute: $x^3 + \frac{1}{x^3} = 2^3 - 3(2) = 8 - 6 = 2$
  4. Answer: $2$

Example 3: Cyclic Sums

Problem: If $x + y + z = 6$ and $xy + yz + zx = 11$, find $x^2 + y^2 + z^2$.

Solution:

  1. Let $s = x + y + z$: $s = 6$
  2. Use identity: $x^2 + y^2 + z^2 = s^2 - 2(xy + yz + zx)$
  3. Substitute: $x^2 + y^2 + z^2 = 6^2 - 2(11) = 36 - 22 = 14$
  4. Answer: $14$

⚠️ Common Pitfalls

Pitfall: Incorrect identity application

  • Fix: Remember $x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2$, not $(x + \frac{1}{x})^2 + 2$
  • Example: If $x + \frac{1}{x} = 3$, then $x^2 + \frac{1}{x^2} = 3^2 - 2 = 7$, not $3^2 + 2 = 11$

Pitfall: Forgetting to check domain

  • Fix: Ensure substitutions are valid
  • Example: $u = x + \frac{1}{x}$ requires $x \neq 0$

Pitfall: Incorrect cyclic substitution

  • Fix: Use correct identity for cyclic sums
  • Example: $x^2 + y^2 + z^2 = (x+y+z)^2 - 2(xy+yz+zx)$, not $(x+y+z)^2$

🎯 AMC-Style Worked Example

Problem: If $x + \frac{1}{x} = 4$, find $x^4 + \frac{1}{x^4}$.

Solution:

  1. Let $u = x + \frac{1}{x}$: $u = 4$
  2. Find $x^2 + \frac{1}{x^2}$: $x^2 + \frac{1}{x^2} = u^2 - 2 = 4^2 - 2 = 16 - 2 = 14$
  3. Use identity: $x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2$
  4. Substitute: $x^4 + \frac{1}{x^4} = 14^2 - 2 = 196 - 2 = 194$
  5. Answer: $194$

Key insight: Higher powers can be found by building up from lower powers.

🔗 Related Patterns

  • Symmetric Expressions — Look for patterns involving $x$ and $\frac{1}{x}$
  • Cyclic Sums — Expressions with $x + y + z$ and related terms
  • Identities — Memorize common symmetry identities
  • Substitution — Replace complex expressions with simpler variables

📝 Practice Checklist

  • Master basic symmetry substitutions
  • Practice higher power identities
  • Learn cyclic sum techniques
  • Practice domain restrictions
  • Understand identity applications
  • Work on speed and accuracy

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