Quadratic Functions

A quadratic function is a function of the form

It can be expressed in the vertex form

by completing the square. The graph of a quadratic function in the vertex form is a parabola with vertex $(h,k)$. If $a>0$, then the quadratic function $f$ has the minimum value $k$ at $x=h=-b/(2a)$. If $a<0$, then the quadratic function $f$ has the maximum value $k$ at $x=h=-b/(2a)$.

Polynomial Functions | Section 3.2 A polynomial function of degree $n$ is a function $P$ of the form

(where $a_n\ne 0$ ). The numbers $a_i$ are the coefficients of the polynomial; $a_n$ is the leading coefficient, and $a_0$ is the constant coefficient (or constant term).

The graph of a polynomial function is a smooth, continuous curve.

Real Zeros of Polynomials

A zero of a polynomial $P$ is a number $c$ for which $P(c)=0$. The following are equivalent ways of describing real zeros of polynomials:

1. $c$ is a real zero of $P$.
2. $x=c$ is a solution of the equation $P(x)=0$.
3. $x-c$ is a factor of $P(x)$.
4. $c$ is an $x$-intercept of the graph of $P$.

Multiplicity of a Zero

A zero $c$ of a polynomial $P$ has multiplicity $m$ if $m$ is the highest power for which $(x-c{)}^m$ is a factor of $P(x)$.

Local Maximums and Minimums

A polynomial function $P$ of degree $n$ has $n-1$ or fewer local extrema (i.e., local maximums and minimums).

Division of Polynomials

If $P$ and $D$ are any polynomials [with $D(x)\ne 0$ ], then we can divide $P$ by $D$ using either long division or synthetic division. The result of the division can be expressed in either of the following equivalent forms:

In this division, $P$ is the dividend, $D$ is the divisor, $Q$ is the quotient, and $R$ is the remainder. When the division is continued to its completion, the degree of $R$ is less than the degree of $D[$ or $R(x)=0]$.

Remainder Theorem

When $P(x)$ is divided by the linear divisor $D(x)=x-c$, the remainder is the constant $P(c)$. So one way to evaluate a polynomial function $P$ at $c$ is to use synthetic division to divide $P(x)$ by $x-c$ and observe the value of the remainder.

Rational Zeros of Polynomials

If the polynomial $P$ given by

has integer coefficients, then all the rational zeros of $P$ have the form

where $p$ is a divisor of the constant term $a_0$ and $q$ is a divisor of the leading coefficient $a_n$. So to find all the rational zeros of a polynomial, we list all the possible rational zeros given by this principle and then check to see which are actual zeros by using synthetic division.

Descartes's Rule of Signs

Let $P$ be a polynomial with real coefficients. Then: The number of positive real zeros of $P$ either is the number of changes of sign in the coefficients of $P(x)$ or is less than that by an even number. The number of negative real zeros of $P$ either is the number of changes of sign in the coefficients of $P(-x)$ or is less than that by an even number.

Upper and Lower Bounds Theorem

Suppose we divide the polynomial $P$ by the linear expression $x-c$ and arrive at the result

If $c>0$ and the coefficients of $Q$, followed by $r$, are all nonnegative, then $c$ is an upper bound for the zeros of $P$.

If $c<0$ and the coefficients of $Q$, followed by $r$ (including zero coefficients), are alternately nonnegative and nonpositive, then $c$ is a lower bound for the zeros of $P$.

The Fundamental Theorem of Algebra, Complete Factorization, and the Zeros Theorem

Every polynomial $P$ of degree $n$ with complex coefficients has exactly $n$ complex zeros, provided that each zero of multiplicity $m$ is counted $m$ times. $P$ factors into $n$ linear factors as follows:

where $a$ is the leading coefficient of $P$ and $c_1,c_1,\dots ,c_n$ are the zeros of $P$.

Conjugate Zeros Theorem

If the polynomial $P$ has real coefficients and if $a+bi$ is a zero of $P$, then its complex conjugate $a-bi$ is also a zero of $P$.

Linear and Quadratic Factors Theorem

Every polynomial with real coefficients can be factored into linear and irreducible quadratic factors with real coefficients.

Rational Functions

A rational function $r$ is a quotient of polynomial functions:

We generally assume that the polynomials $P$ and $Q$ have no factors in common.

Asymptotes

The line $x=a$ is a vertical asymptote of the function $y=f(x)$ if

The line $y=b$ is a horizontal asymptote of the function $y=f(x)$ if

Asymptotes of Rational Functions

Let $r(x)=\frac{P(x)}{Q(x)}$ be a rational function. (continued)

The vertical asymptotes of $r$ are the lines $x=a$ where $a$ is a zero of $Q$. If the degree of $P$ is less than the degree of $Q$, then the horizontal asymptote of $r$ is the line $y=0$.

If the degrees of $P$ and $Q$ are the same, then the horizontal asymptote of $r$ is the line $y=b$, where

If the degree of $P$ is greater than the degree of $Q$, then $r$ has no horizontal asymptote.

Polynomial and Rational Inequalities

A polynomial inequality is an inequality of the form $P(x)\ge 0$, where $P$ is a polynomial. We solve $P(x)\ge 0$ by finding the zeros of $P$ and using test values between successive zeros to determine the intervals that satisfy the inequality. A rational inequality is an inequality of the form $r(x)\ge 0$, where

is a rational function. The cut points of $r$ are the values of $x$ at which either $P(x)=0$ or $Q(x)=0$. We solve $r(x)\ge 0$ by using test points between successive cut points to determine the intervals that satisfy the inequality.