Polynomial and Rational Functions

Zeros of Polynomial Functions

Understanding the zeros of polynomial functions is essential for solving equations and analyzing the behavior of polynomials. This section covers several theorems and techniques for finding and classifying zeros.

The Rational Zero Theorem

• Definition: If a polynomial function with integer coefficients has a rational zero (in lowest terms), then p is a factor of the constant term , and q is a factor of the leading coefficient .

• Application: To list all possible rational zeros, determine all factors of the constant term and the leading coefficient, then form all possible fractions .

• Example: For , the possible rational zeros are all combinations of factors of over factors of $4$.

Finding Zeros of a Polynomial Function

• List all possible rational zeros using the Rational Zero Theorem.

• Test each candidate zero using synthetic division or direct substitution.

• Once a zero is found, factor it out and repeat the process for the reduced polynomial.

• Example: For a cubic polynomial with possible rational zeros , use synthetic division to test each value until a zero is found.

Properties of Roots of Polynomial Equations

• If a polynomial equation is of degree n, then (counting multiplicities) it has n roots.

• If is a root of a polynomial with real coefficients, then its complex conjugate is also a root.

Solving Polynomial Equations

• List all possible rational roots using the Rational Zero Theorem.

• Test each root using synthetic division.

• Once a root is found, factor the polynomial and solve the remaining equation (possibly using the quadratic formula for degree 2).

• Example: For , possible rational roots are .

The Fundamental Theorem of Algebra

• If is a polynomial of degree , then the equation has at least one complex root.

The Linear Factorization Theorem

• Any th-degree polynomial with complex coefficients can be written as the product of a nonzero constant and linear factors:

• Where are complex numbers (possibly real and not necessarily distinct).

• Example: If a polynomial has zeros at and , and , use the theorem to construct the polynomial, remembering to include the conjugate if coefficients are real.

Descartes’ Rule of Signs

• Positive Real Zeros: The number of positive real zeros of is either equal to the number of sign changes in or less than that by an even integer.

• Negative Real Zeros: The number of negative real zeros of is either equal to the number of sign changes in or less than that by an even integer.

• Example: If has 4 sign changes, possible numbers of positive real zeros are 4, 2, or 0. If has no sign changes, there are no negative real zeros.