Quadratic Functions

A quadratic function is a function of the form

\[ f(x)=a x^{2}+b x+c \]

It can be expressed in the vertex form

\[ f(x)=a(x-h)^{2}+k \]

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 \(x=h=-b /(2 a)\). If \(a<0\), then the quadratic function \(f\) has the maximum value \(k\) at \(x=h=-b /(2 a)\).

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

\[ P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \]

(where \(a_{n} \neq 0\) ). 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 \(D(x) \neq 0\) ], 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:

\[ \begin{aligned} P(x) & =D(x) \cdot Q(x)+R(x) \\ \frac{P(x)}{D(x)} & =Q(x)+\frac{R(x)}{D(x)} \end{aligned} \]

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

\[ P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \]

has integer coefficients, then all the rational zeros of \(P\) have the form

\[ x= \pm \frac{p}{q} \]

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

\[ P(x)=(x-c) \cdot Q(x)+r \]

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:

\[ P(x)=a\left(x-c_{1}\right)\left(x-c_{2}\right) \cdots\left(x-c_{n}\right) \]

where \(a\) is the leading coefficient of \(P\) and \(c_{1}, c_{1}, \ldots, c_{n}\) are the zeros of \(P\).

Conjugate Zeros Theorem

If the polynomial \(P\) has real coefficients and if \(a+b i\) is a zero of \(P\), then its complex conjugate \(a-b i\) 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:

\[ r(x)=\frac{P(x)}{Q(x)} \]

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

\[ y \rightarrow \infty \quad \text { or } \quad y \rightarrow-\infty \quad \text { as } \quad x \rightarrow a^{+} \quad \text { or } \quad x \rightarrow a^{-} \]

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

\[ y \rightarrow b \quad \text { as } \quad x \rightarrow \infty \quad \text { or } \quad x \rightarrow-\infty \]

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

\[ b=\frac{\text { leading coefficient of } P}{\text { leading coefficient of } Q} \]

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 \(P(x) \geq 0\), where \(P\) is a polynomial. We solve \(P(x) \geq 0\) 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 \(r(x) \geq 0\), where

\[ r(x)=\frac{P(x)}{Q(x)} \]

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 \(r(x) \geq 0\) by using test points between successive cut points to determine the intervals that satisfy the inequality.