Finding Zeros of a Polynomial

INTRODUCTION AND GOAL

Finding the zeros of a polynomial means solving the equation P(x) = 0, where P(x) is a polynomial with real or rational coefficients. The goal is to determine all values of x for which the polynomial has a value of 0. These values are key for factorization, graphical representation, and further algebraic procedures such as division and simplification of expressions.

A polynomial of degree 'n' has at most 'n' real or complex zeros, counting multiplicities.

METHODS FOR FINDING ZEROS

1. FACTORING OUT THE GREATEST COMMON FACTOR (GCF)
   If all terms have a common factor, first factor it out to simplify the polynomial.
   Example: P(x) = x³ – x² = x²(x – 1) → Zeros: x = 0 (with multiplicity 2), x = 1.
2. FACTORING THE POLYNOMIAL
   If the polynomial can be rewritten as a product of lower-degree factors, the zeros can be found from each factor.
   Example: P(x) = (x – 2)(x + 3)(x – 1) → Zeros: x = 2, x = –3, x = 1.
3. RATIONAL ROOT THEOREM (FOR POLYNOMIALS WITH INTEGER COEFFICIENTS)
   If a polynomial with integer coefficients has rational zeros, they are of the form:
   x = ±p/q,
   where 'p' is a divisor of the constant term (a₀), and 'q' is a divisor of the leading coefficient (aₙ).
   Example: P(x) = x³ – 2x² – 5x + 6
   Possible rational zeros (divisors of 6 divided by divisors of 1): ±1, ±2, ±3, ±6.
   By testing: P(1) = 1³ – 2(1)² – 5(1) + 6 = 1 – 2 – 5 + 6 = 0 → x = 1 is a zero.
4. HORNER'S ALGORITHM (SYNTHETIC DIVISION)
   Once one zero 'c' is known, the polynomial can be divided by (x – c) using Horner's algorithm (or synthetic division) to find the remaining quotient, whose zeros can then be sought.
   Example: P(x) = x³ – 6x² + 11x – 6
   Test P(1) = 1³ – 6(1)² + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 → x = 1 is a zero.
   Divide P(x) by (x – 1):
   The quotient is x² – 5x + 6.
   Factoring the quadratic: x² – 5x + 6 = (x-2)(x-3) → Zeros of the quotient are x = 2, x = 3.
   Final zeros: x = 1, x = 2, x = 3.
5. FORMULAS FOR SPECIAL CASES
6. • QUADRATIC POLYNOMIAL: P(x) = ax² + bx + c → Zeros using the quadratic formula:
     x = (–b ± √(b² – 4ac)) / 2a
   • CUBIC OR HIGHER-DEGREE POLYNOMIALS: If factoring or the rational root theorem is not straightforward, numerical methods or more advanced techniques might be required.
6. GRAPHICAL METHOD
   Zeros can also be approximated by looking at the graph of the function f(x) = P(x) – the points where the graph intersects the x-axis correspond to the real zeros.

CONCLUSION

Finding the zeros of a polynomial is a systematic process based on factoring, testing rational values, division, and using known formulas. Knowledge of methods for finding zeros allows for understanding the structure of a polynomial, its factorization, and preparation for solving more complex equations.