Higher-Degree Equations

Substitution Method

Reduce a higher-degree equation to a quadratic by substituting a new variable. For a biquadratic equation ax⁴ + bx² + c = 0, let t = x² to get at² + bt + c = 0. Solve for t, then x = ±√t.

Example: x⁴ − 5x² + 4 = 0: let t = x², then t² − 5t + 4 = 0 → t = 1 or 4 → x = ±1, ±2.

Factorization

If a polynomial can be factored, set each factor to zero. The Rational Root Theorem: any rational root p/q must have p | a₀ and q | aₙ.

Symmetric (Reciprocal) Equations

Equations where coefficients are symmetric: ax⁴ + bx³ + cx² + bx + a = 0. Divide by x² and substitute t = x + 1/x. The equation becomes quadratic in t.

Example: 6x⁴ − 5x³ − 38x² − 5x + 6 = 0. Divide by x², substitute t = x + 1/x: 6t² − 5t − 50 = 0. Solutions: x = −2, −1/2, 1/3, 3.

Method of Symmetrization

Equations of the form (x + a)ⁿ + (x + b)ⁿ = c are solved using t = x + (a + b)/2.

Example: (x + 3)⁴ + (x + 1)⁴ = 272 → substitute t = x + 2 → 2t⁴ + 48t² = 0 → x = 1 or −5.