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Boundless Algebra
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Chapter 7

Polynomials and Rational Functions

Book Version 13
By Boundless
Boundless Algebra
Algebra
by Boundless
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Section 1
Introduction to Polynomials
What Are Polynomials?

A polynomial is a finite expression containing constants and variables connected only through basic operations of algebra.

Adding and Subtracting Polynomials

Polynomials can be added or subtracted by combining like terms.

Multiplying Polynomials

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

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Polynomial and Rational Functions as Models

Functions are commonly used in fitting data to a trend line. Polynomial and rational functions are both relatively accurate and easy to use.

Section 2
Graphing Polynomial Functions
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Basics of Graphing Polynomial Functions

A polynomial function in one real variable can be represented by a graph.

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The Leading-Term Test

Analysis of a polynomial reveals whether the function will increase or decrease as $x$ approaches positive and negative infinity.

Section 3
Polynomial Division
Introduction to Factoring Polynomials

Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.

Division and Factors

Polynomial long division functions similarly to long division, and if the division leaves no remainder, then the divisor is called a factor.

The Remainder Theorem and Synthetic Division

Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder.

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Finding Factors of Polynomials

Finding factors of polynomials is important, since it is always best to work with the simplest version of a polynomial.

Section 4
Zeroes of Polynomial Functions
The Fundamental Theorem of Algebra

The fundamental theorem states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

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Finding Polynomials with Given Zeros

To construct a polynomial from given zeros, set $x$ equal to each zero, move everything to one side, then multiply each resulting equation.

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Finding Zeros of Factored Polynomials

The factored form of a polynomial reveals its zeros, which are defined as points where the function touches the $x$-axis.

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Integer Coefficients and the Rational Zeros Theorem

Each solution to a polynomial, expressed as $x= \frac {p}{q}$, must satisfy that $p$ and $q$ are integer factors of $a_0$ and $a_n$, respectively.

The Rule of Signs

The rule of signs gives an upper bound number of positive or negative roots of a polynomial.

Section 5
Rational Functions
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Introduction to Rational Functions

A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.

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Asymptotes

A rational function can have at most one horizontal or oblique asymptote, and many possible vertical asymptotes; these can be calculated.

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Solving Problems with Rational Functions

The $x$-intercepts of rational functions are found by setting the polynomial in the numerator equal to $0$ and solving for $x$.

Simplifying, Multiplying, and Dividing

A rational expression can be treated like a fraction, and can be manipulated via multiplication and division.

Partial Fractions

Partial fraction decomposition is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.

Section 6
Inequalities With Polynomial and Rational Functions
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Polynomial Inequalities

Polynomials can be expressed as inequalities, the solutions for which can be determined from the polynomial's zeros.

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Rational Inequalities

Rational inequalities can be solved much like polynomial inequalities.

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Polynomials and Rational Functions
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