Download a PDF of this page here.

Download the Spanish version here.

Essential Concepts

Introduction to Power and Polynomial Functions

• A power function is a variable base raised to a number power.
• The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
• The end behavior depends on whether the power is even or odd.
• A polynomial function is the sum of terms, each of which consists of a transformed power function with non-negative integer powers.
• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
• The sign of the leading term will determine the direction of the ends of the graph:
  • even degree and positive coefficient: both ends point up
  • even degree and negative coefficient: both ends point down
  • odd degree and positive coefficient: the left-most end points down and the right-most end points up.
  • odd degree and negative coefficient: the left-most end points up and the right-most end points down.
• The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.
• A polynomial of degree [latex]n[/latex] will have at most [latex]n[/latex] [latex]x[/latex]-intercepts and at most [latex]n – 1[/latex] turning points.

Graphs of Polynomial Functions

• Polynomial functions of degree 2 or more are smooth, continuous functions.
• To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
• Another way to find the [latex]x[/latex]–intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the [latex]x[/latex]-axis.
• The multiplicity of a zero determines how the graph behaves at the [latex]x[/latex]-intercept.
  • The graph of a polynomial will cross the [latex]x[/latex]-axis at a zero with odd multiplicity.
  • The graph of a polynomial will touch and bounce off the [latex]x[/latex]-axis at a zero with even multiplicity.
• The graph of a polynomial will cross the [latex]x[/latex]-axis at a zero with odd multiplicity.
• The graph of a polynomial will touch and bounce off the [latex]x[/latex]-axis at a zero with even multiplicity.
• The end behavior of a polynomial function depends on the leading term.
• The graph of a polynomial function changes direction at its turning points.
• To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most [latex]n – 1[/latex] turning points.
• Graphing a polynomial function helps to estimate local and global extremas.
• The Intermediate Value Theorem tells us that if [latex]f\left(a\right) \text{and} f\left(b\right)[/latex] have opposite signs, then there exists at least one value [latex]c[/latex] between [latex]a[/latex] and [latex]b[/latex] for which [latex]f\left(c\right)=0[/latex].

Dividing Polynomials

• Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.
• The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
• Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form [latex]x – k[/latex].
• Polynomial division can be used to solve application problems, including area and volume.

Zeros of Polynomial Functions

• To find [latex]f\left(k\right)[/latex], determine the remainder of the polynomial [latex]f\left(x\right)[/latex] when it is divided by [latex]x-k[/latex].
• [latex]k[/latex] is a zero of [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex]  is a factor of [latex]f\left(x\right)[/latex].
• Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.
• When the leading coefficient is [latex]1[/latex], the possible rational zeros are the factors of the constant term.
• Synthetic division can be used to find the zeros of a polynomial function.
• According to the Fundamental Theorem of Algebra, every polynomial function has at least one complex zero.
• Every polynomial function with degree greater than [latex]0[/latex] has at least one complex zero.
• Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form [latex]\left(x-c\right)[/latex] where c is a complex number.
• The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
• The number of negative real zeros of a polynomial function is either the number of sign changes of [latex]f\left(-x\right)[/latex]  or less than the number of sign changes by an even integer.
• Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.
• Remainder Theorem states that if a polynomial [latex]f\left(x\right)[/latex] is divided by [latex]x-k[/latex] , then the remainder is equal to the value [latex]f\left(k\right)[/latex].
• Rational Zero Theorem states that the possible rational zeros of a polynomial function have the form [latex]\frac{p}{q}[/latex] where [latex]p[/latex] is a factor of the constant term and [latex]q[/latex] is a factor of the leading coefficient.
• Factor Theorem states that [latex]k[/latex] is a zero of polynomial function [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex]  is a factor of [latex]f\left(x\right)[/latex].
• The Fundamental Theorem of Algebra states that a polynomial function with degree greater than [latex]0[/latex] has at least one complex zero.
• Linear Factorization Theorem states that a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\left(x-c\right)[/latex] where c is a complex number.
• Complex Conjugates Theorem states that if the polynomial function [latex]f[/latex] has real coefficients and a complex zero of the form [latex]a+bi[/latex], then the complex conjugate of the zero, [latex]a−bi[/latex], is also a zero.
• Descartes’ Rule of Signs is a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\left(x\right)[/latex] and [latex]f\left(-x\right)[/latex].

Key Equations

general form of a polynomial function	[latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
Division Algorithm	[latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex] where [latex]q\left(x\right)\ne 0[/latex]

Glossary

coefficient

a nonzero real number multiplied by a variable raised to an exponent

continuous function

a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

degree

the highest power of the variable that occurs in a polynomial

Descartes’ Rule of Signs

a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\left(x\right)[/latex] and [latex]f\left(-x\right)[/latex]

Division Algorithm

given a polynomial dividend [latex]f\left(x\right)[/latex] and a non-zero polynomial divisor [latex]d\left(x\right)[/latex] where the degree of [latex]d\left(x\right)[/latex] is less than or equal to the degree of [latex]f\left(x\right)[/latex], there exist unique polynomials [latex]q\left(x\right)[/latex] and [latex]r\left(x\right)[/latex] such that [latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex] where [latex]q\left(x\right)[/latex] is the quotient and [latex]r\left(x\right)[/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\left(x\right)[/latex].

end behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

Factor Theorem

k is a zero of polynomial function [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex]  is a factor of [latex]f\left(x\right)[/latex]

Fundamental Theorem of Algebra

a polynomial function with degree greater than 0 has at least one complex zero

global maximum

highest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all [latex]x[/latex].

global minimum

lowest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\le f\left(x\right)[/latex] for all [latex]x[/latex].

Intermediate Value Theorem

for two numbers [latex]a[/latex] and [latex]b[/latex] in the domain of [latex]f[/latex], if [latex]a

leading coefficient

the coefficient of the leading term

leading term

the term containing the highest power of the variable

Linear Factorization Theorem

allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\left(x-c\right)[/latex] where c is a complex number

multiplicity

the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], [latex]x=h[/latex] is a zero of multiplicity [latex]p[/latex].

polynomial function

a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

power function

a function that can be represented in the form [latex]f\left(x\right)=a{x}^{n}[/latex] where a is a constant, the base is a variable, and the exponent is n, is a smooth curve represented by a graph with no sharp corners

Rational Zero Theorem

the possible rational zeros of a polynomial function have the form [latex]\frac{p}{q}[/latex] where p is a factor of the constant term and q is a factor of the leading coefficient

Remainder Theorem

if a polynomial [latex]f\left(x\right)[/latex] is divided by [latex]x-k[/latex] , then the remainder is equal to the value [latex]f\left(k\right)[/latex]

synthetic division

a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x – k[/latex]

term of a polynomial function

any [latex]{a}_{i}{x}^{i}[/latex] of a polynomial function in the form [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

turning point

the location where the graph of a function changes direction