ZEROS OF A POLYNOMIAL

DEFINITION AND MEANING

The zeros of a polynomial are those values of the variable x for which the polynomial is equal to zero. If P(x) is a given polynomial, then x₀ is its zero if:

P(x₀) = 0

Solving the equation P(x) = 0 means finding all the zeros of the polynomial. These values play a key role in factorization, graphical representation, and the analysis of functions, as they indicate the points where the graph of the polynomial intersects the x-axis.

RELATIONSHIP BETWEEN ZEROS AND FACTORS

If x₀ is a zero of a polynomial, then (x – x₀) is a divisor (factor) of that polynomial. Conversely, if a polynomial has a factor (x – c), then c is its zero. This is known as the Factor Theorem.

Example:
If P(x) = (x – 2)(x + 1), then its zeros are x = 2 and x = –1.

NUMBER OF ZEROS

A polynomial of degree 'n' (where n ≥ 1) has at most 'n' real zeros, and exactly 'n' complex zeros if we count them with multiplicity. Multiplicity indicates how many times a particular zero is repeated as a solution.

For example:
P(x) = (x – 3)²(x + 2) has:

• a zero at 3 with multiplicity 2,
• a zero at –2 with multiplicity 1.

REAL AND COMPLEX ZEROS

• A polynomial with real coefficients can have real or complex zeros.
• If a polynomial has real coefficients, any complex zeros always occur in conjugate pairs (e.g., if z is a zero, then its conjugate, conjugate(z), is also a zero).

METHODS FOR FINDING ZEROS

• Solving the equation P(x) = 0 directly or using formulas (like the quadratic formula for degree 2).
• Factoring the polynomial into simpler factors.
• Using Horner's algorithm (synthetic division) for dividing the polynomial and testing possible zeros.
• Using the Rational Root Theorem – if all coefficients are integers, rational zeros are sought among the divisors of the constant term divided by the divisors of the leading coefficient.

EXAMPLE

Let P(x) = x³ – 6x² + 11x – 6. We are looking for its zeros:

1. Using the Rational Root Theorem:
   Possible rational candidates (divisors of -6 divided by divisors of 1): ±1, ±2, ±3, ±6.
2. Test candidates:
   P(1) = 1³ – 6(1)² + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 → x = 1 is a zero.
3. Divide P(x) by (x – 1) using synthetic division or long division. The quotient will be x² – 5x + 6.
4. Find zeros of the quotient:
   x² – 5x + 6 = 0
   This can be factored as (x – 2)(x – 3) = 0.
   So, the remaining zeros are x = 2 and x = 3.

Final zeros: x = 1, x = 2, x = 3.

GRAPHICAL MEANING

Each real zero of a polynomial is the x-coordinate of an intersection point of the graph of the function f(x) = P(x) with the x-axis.

• If a zero has an even multiplicity, the graph touches the x-axis at that zero (and turns around).
• If a zero has an odd multiplicity, the graph crosses the x-axis at that zero.

CONCLUSION

The zeros of a polynomial are a fundamental element in understanding and working with polynomial functions. With their help, we can factor polynomials, analyze the course of their graphs, and solve equations. The connection between zeros and factors allows for a clear decomposition of even more complex expressions.