Zeros of Polynomial Functions – College Algebra Study Notes

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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 using synthetic division.

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 (where ) and the polynomial has real coefficients, then its complex conjugate is also a root.
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 (real or complex) of .
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.
Example: Zeros yield .

Descartes’s Rule of Signs

Used to determine the possible number of positive and negative real zeros of a polynomial function.
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.
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 and Theorems for Polynomial Zeros

Theorem/Property	Main Idea	Application
Rational Zero Theorem	Lists all possible rational zeros	Testing zeros using synthetic division
Fundamental Theorem of Algebra	Degree polynomial has $n$ roots (complex)	Counting total roots
Linear Factorization Theorem	Polynomial can be factored into linear factors	Constructing polynomials from zeros
Descartes’s Rule of Signs	Predicts number of positive/negative real zeros	Analyzing sign changes in and

Additional info: These notes expand on the brief points in the original material, providing definitions, examples, and context for each theorem and property. The image referenced was not included, as it was not clearly and directly relevant to the explanation of any specific paragraph.