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Polynomial Roots

Find Polynomial Roots

Finding Polynomial Roots

The roots of a polynomial are the values of x that make the polynomial equal to zero. Finding these roots is fundamental in algebra and has applications across mathematics and science. Here are the common methods:

  1. Quadratic Formula

    For quadratic polynomials (degree 2), use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) where a, b, and c are the coefficients of ax² + bx + c.

  2. Factorization

    For polynomials that can be factored, break them down into simpler polynomials whose roots can be easily found. This works well when the polynomial has rational roots.

  3. Rational Root Theorem

    This theorem helps identify possible rational roots by testing factors of the constant term divided by factors of the leading coefficient.

  4. Numerical Methods

    When analytical solutions are difficult, numerical methods like Newton's method or the bisection method can approximate roots to any desired accuracy.

Example: Finding Roots of x³ - 6x² + 11x - 6

1. Try Rational Root Theorem candidates: ±1, ±2, ±3, ±6 2. Test x = 1: (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 → Root found! 3. Factor out (x - 1): x³ - 6x² + 11x - 6 = (x - 1)(x² - 5x + 6) 4. Factor the quadratic: x² - 5x + 6 = (x - 2)(x - 3) 5. Final roots: x = 1, x = 2, x = 3