Polynomial Functions (College Board AP® Precalculus): Study Guide

Polynomial functions

What is a polynomial function?

• A polynomial function of $x$ is any function that can be written in the form $p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\dots +a_2{x}^2+a_{1}x+a_0$

  • where $n$ is a positive integer

  • and $a_i$ is a real number for each $i$ from 0 to $n$, with $a_n\ne 0$

What are the key terms for polynomial functions?

• The degree of a polynomial is the highest power of $x$ that appears in the expression

  • E.g. $p\left(x\right)=3x^4-2x^2+x-7$ has degree $4$

• The leading term is the term with the highest power of $x$

  • E.g. for $p\left(x\right)=3x^4-2x^2+x-7$, the leading term is $3x^4$

• The leading coefficient is the coefficient of the leading term

  • E.g. for $p\left(x\right)=3x^4-2x^2+x-7$, the leading coefficient is $3$

• These three properties (degree, leading term, and leading coefficient) are important

  • They determine much of the polynomial's behavior

    • including its end behavior and general shape

How do familiar function types relate to polynomials?

• Many function types that you already know are special cases of polynomial functions:

  • A constant function, such as $f\left(x\right)=5$, is a polynomial of degree 0

    • It can be thought of as $f\left(x\right)=5x^0=5$

  • A linear function, such as $f\left(x\right)=2x+3$, is a polynomial of degree 1

  • A quadratic function, such as $f\left(x\right)=-x^2+4x-1$, is a polynomial of degree 2

• Polynomials of higher degree include:

  • Cubic functions (degree 3), e.g. $p\left(x\right)=2x^3-x+4$

  • Quartic functions (degree 4), e.g. $p\left(x\right)=x^4-3x^3+2x$

• There is no limit to the maximum degree a polynomial function can have

How can I recognize a polynomial function?

• A polynomial function must be a sum (or difference) of terms

  • where each term consists of a constant multiplied by a non-negative integer power of $x$

  • The constant may be 1

    • e.g. $x^3=1x^3$

• The following are not polynomial functions:

  • $f\left(x\right)=\frac{1}{x}$

    • this involves a negative power of $x$ (i.e. $x^{-1}$)

  • $f\left(x\right)=\sqrt{x}$

    • this involves a fractional power of $x$ (i.e. $x^{1/2}$)

  • $f\left(x\right)=2^x$

    • this is an exponential function, not a polynomial

  • $f\left(x\right)=\sin \left(x\right)$

    • this is a trigonometric function, not a polynomial

Does the order of terms matter?

• A polynomial does not have to be written with the terms in order of decreasing degree (or in any particular order)

  • E.g. $p\left(x\right)=1-3x+5x^3$ is the same polynomial as $p\left(x\right)=5x^3-3x+1$

  • The degree is still $3$, the leading term is still $5x^3$, and the leading coefficient is still $5$

• To identify the degree and leading term, look for the term with the highest power of $x$, regardless of where it appears in the expression