**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,a_{0} 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 a_{0} 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^{2}+b^{2} (
__real__!) ; z_{1}/z_{2} = (z_{1}·[`(z_{2})])/(z_{2}·[`(z_{2})])

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_{n}x^{n}+¼+a_{0}, q(x) = b_{m}x^{m}+¼+b_{0} ; 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^{loga 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^{logb 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

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