Solving Higher Degree Equations Using Factoring

Study Notes

Many higher degree equations can be solved by factoring them into simpler expressions.
This allows you to find solutions by setting each factor equal to zero and solving for x.
The key is to identify patterns and use algebraic manipulations to factor the equation.

Grouping: Look for common factors within groups of terms.
Difference of Squares: Recognize expressions in the form (a² - b²) which can be factored as (a + b)(a - b).
Sum/Difference of Cubes: Identify expressions in the form (a³ + b³) or (a³ - b³) which can be factored using specific formulas.
Substitution: Introduce a new variable (e.g., let y = x²) to simplify the equation and make it easier to factor.

Example 1: x⁴ - 10x² - x + 20 = 0
- Factoring by grouping:
  - Group the first two terms and the last two terms:
    - (x⁴ - 10x²) - (x - 20) = 0
  - Factor out common factors:
    - x²(x² - 10) - 1(x - 20) = 0
  - Notice that (x² - 10) and (x - 20) can't be further factored.
  - This equation does not factor easily, and may require other methods to solve.
Example 2: (x + 2)(x - 3)(x + 4)(x - 6) + 6x² = 0
- Factoring a difference of squares:
  - Notice that the first four terms represent a product of four factors, and the last term is a perfect square.
  - By rearranging and factoring, it can be rewritten as:
    - [(x + 2)(x - 6)] [(x - 3)(x + 4)] + 6x² = 0
    - (x² - 4x - 12)(x² + x - 12) + 6x² = 0
  - Identify a difference of squares:
    - (x² - 4x - 12 + √6x)(x² - 4x - 12 - √6x) = 0
  - Factor each quadratic expression separately:
    - (x - 2 + √6)(x - 2 -√6)(x - 6)(x + 2) = 0
    - Setting each factor equal to zero leads to the solutions for x.
Example 3: 3(x² - x + 1)² - 2(x + 1)² = 5(x³ + 1)
- Factoring a sum of cubes:
  - Simplify and rewrite the expression as:
    - 3(x² - x + 1)² - 2(x + 1)² - 5(x + 1)(x² - x + 1) = 0
  - Notice that (x² - x + 1) appears multiple times.
  - Let y = (x² - x + 1):
    - 3y² - 2(x + 1)² - 5(x + 1)y = 0
  - This is now a quadratic equation in y. Solve for y, and then substitute back to find the solutions for x.

Not all higher degree equations are solvable by factoring.
Always try to simplify and rearrange terms before attempting to factor.
Look for patterns and commonalities in the expression.
Be aware of factoring methods specific to certain forms of equations.
Practice with various examples to develop your problem-solving skills.
Consider other methods, such as the rational root theorem or numerical methods, when factoring proves difficult.