Chapter 2: Polynomials

§ 3:

Polynomial division

reason: polynomial (long) division

f(x) = (x-a)g(x) + b ; a=root, then b=0

polonomial = (divisor)(quotient) + remainder

degree of remainder is less than degree of divisor

synthetic division: fast method to divide by (x-a)

§ 4:

Real zeros of polynomial functions

`Counting' zeros of f

Descartes' rule of signs

p=number of positive roots of f, q=number of negative roots of f

(number of changes in sign of coeffs of f) - p is ³ 0 and even

(number of changes in sign of coeffs of f(-x)) - q is ³ 0 and even

Rational roots test

If a_n,¼,a₀ are all integers, a_n ¹ 0, and r = p/q is a rational root of f, then

q divides a_n evenly and p divides a₀ evenly.

backwards: can show roots of a polynomial can't be rational.

Bounding roots: start with a_n > 0.

If c > 0 and the bottom row after synthetic division of f using c are all ³ 0,

then no root of f is bigger than c.

If c < 0 and the bottom alternates sign, then no root of f is smaller than c.

§ 5:

Complex numbers

i = [Ö(-1)], pretend i behaves like a real number

complex numbers: standard form z = a+bi ; addition, subtraction, multiplication

division: complex conjugate [`z] = a-bi

z·[`z] = a<sup>2</sup>+b<sup>2</sup> ( real!) ; z<sub>1</sub>/z<sub>2</sub> = (z<sub>1</sub>·[`(z₂)])/(z₂·[`(z₂)])

a,b > 0, then [Ö(-a)]·[Ö(-b)] = -[Ö((-a)(-b))] (unfortunately)

§ 6:

The fundamental theorem of algebra

complex root r ; f(r)=0

Every polynomial factors into linear factors (with coefficients in C)

FTA says it can be done; it doesn't tell you how to do it!

Conjugate pairs; if coeffs of f are real, and r is a root, then so is [`r]

(x-r)(x-[`r]) has real coeffs

every polynomial with real coeffs factors in linear and irreducible quadratic factors.

§ 7:

Rational functions

p(x) = a_nxⁿ+¼+a₀, q(x) = b_mx^m+¼+b₀ ; f(x) = p(x)/q(x)

domain = where q(x) ¹ 0

vertical asymptote x = a : f(x)®±¥ as x®a

horizontal asymptote: f(x)®a as x®±¥

n < m : horiz. asymp. y = 0

n = m : horiz. asymp. y = a_n/b_m

n > m : no horiz. asymp.

Slant asymptote: n = m+1 . Asymp. = linear part from division of p(x) by q(x)

Chapter 3: Exponential and logarithmic functions

§ 1:

Exponential functions

Rules: a^b+c = a^b a^c ; a^bc = (a^b)^c ; (ab)^c = a^c b^c

Function f(x) = a^x ; approximate f(x) by f(rational number close to x)

Domain: R ; range: (0,¥) ; horiz. asymp. y = 0

Graphs:

a > 10 < a < 1

Most natural base: e = 2.718281829459045.....

Exponential growth: compound interest

P=initial amount, r=interest rate, compounded n times/year

A(t) = P·(1+r/n)^nt

n®¥, continuous compounding : A(t) = Pe^rt

Radioactive decay: half-life = k (A(k) = A(0)/2)

A(t) = A(0)(1/2)^t/k

§ 2:

Logarithmic functions

log_a x is the inverse of a^x

a = base of the logarithm

log_a (a^x) = x, all x ; a^{log_a x} = x, all x > 0

Domain: all x > 0 ; range: all x

Graph = reflection of graph of a^x across line y = x

vertical asymptote: x = 0

natural logarithm: log_e x = lnx

§ 3:

Properties of logarithms

log_a (bc) = log_a b + log_a c ; log_a (b^c) = clog_a b

(log_b c)(log_a b) = log_a (b^{log_b c}) = log_a c; so log_b c = [(log_a c)/(log_a b)]

E.g., a = e : log_b c = [lnc/lnb]

§ 4:

Exponential and logarithmic equations

a^blah = bleh, then (blah)lna = ln(bleh)

(2^x-3)(2^x-7) = 0, then 2^x=3 or 2^x=7

logarithmic equation: combine into a single log (one on each side?) and

exponentiate both sides

Application: doubling time

time for investment to triple at interest rate of r compounded n times/year:

solve (1+r/n)^nt = 3