Polynomial and Rational Functions

Zeros of Polynomial Functions

This section explores methods for finding the zeros (roots) of polynomial functions, including the Rational Zero Theorem, synthetic division, properties of roots, and Descartes’s Rule of Signs. Understanding these concepts is essential for solving polynomial equations and analyzing their graphs.

The Rational Zero Theorem

• Definition: The Rational Zero Theorem provides a list of possible rational zeros for a polynomial function with integer coefficients.

• Statement: If a polynomial has integer coefficients, then every rational zero (in lowest terms) satisfies:

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

• Example: For , possible rational zeros are .

Finding Zeros of a Polynomial Function

• Test each possible rational zero using synthetic division or direct substitution.

• If the remainder is zero, the tested value is a zero of the polynomial.

• Continue testing until all zeros are found or the polynomial is factored completely.

• Example: For , possible rational zeros are . Test each value to find the actual zeros.

Properties of Roots of Polynomial Equations

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

• If is a root of a polynomial with real coefficients (where ), then its complex conjugate is also a root. Imaginary roots occur in conjugate pairs.

• Example: If is a root, then is also a root.

Solving Polynomial Equations

• Set the polynomial equal to zero and solve for using factoring, synthetic division, or the Rational Zero Theorem.

• Repeated roots (multiplicity) occur when a factor appears more than once.

• Example: has as a repeated root.

The Fundamental Theorem of Algebra

• Every nonzero polynomial equation of degree with complex coefficients has exactly $n$ roots in the complex number system (counting multiplicities).

The Linear Factorization Theorem

• If is a polynomial of degree , then $f(x)$ can be factored as:

• where are the roots of (real or complex).

• Example: If zeros are , then .

Finding a Polynomial Function with Given Zeros

• Given zeros , construct the polynomial as , where is a nonzero constant.

• Expand the product to write the polynomial in standard form if required.

Descartes’s Rule of Signs

• Used to determine the possible number of positive and negative real zeros of a polynomial function.

• Rule for Positive Real Zeros: The number of positive real zeros of is equal to the number of sign changes in $f(x)$, or less than that by an even integer.

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

• If has only one sign change, then has exactly one negative real zero.

• Example: For , has three sign changes, so there are 3 or 1 positive real zeros.

Summary Table: Properties of Roots and Theorems