Math 1650

Topics for second exam

(Technically, everything covered on the first exam, plus...)

Chapter 2: Polynomials

§ 3:
Polynomial division
root a of f « factor (x-a) of f(x)

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
f(x) = anxn+¼+a1x+a0

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

q divides an evenly and p divides a0 evenly.

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

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
Some polynomials have no roots, e.g., f(x) = x2+1 . Invent some!

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] = a2+b2 ( real!) ; z1/z2 = (z1·[`(z2)])/(z2·[`(z2)])

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

§ 6:
The fundamental theorem of algebra
FTA: Every polynomial f(x) (with coefficients in C or R) has a

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
rational function = quotient of polynomials

p(x) = anxn+¼+a0, q(x) = bmxm+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 = an/bm

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
exponential expressions ab

Rules: ab+c = ab ac ; abc = (ab)c ; (ab)c = ac bc

Function f(x) = ax ; 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) = Pert

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

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

§ 2:
Logarithmic functions
loga x = the number you raise a to to get x

loga x is the inverse of ax

a = base of the logarithm

loga (ax) = x, all x ; aloga x = x, all x > 0

Domain: all x > 0 ; range: all x

Graph = reflection of graph of ax across line y = x

vertical asymptote: x = 0

natural logarithm: loge x = lnx

§ 3:
Properties of logarithms
logarithms undo exponentials; properties are `reverse' of exponentials

loga (bc) = loga b + loga c ; loga (bc) = cloga b

(logb c)(loga b) = loga (blogb c) = loga c; so logb c = [(loga c)/(loga b)]

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

§ 4:
Exponential and logarithmic equations
exponential equation: take logs!

ablah = bleh, then (blah)lna = ln(bleh)

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

File translated from TEX by TTH, version 0.9.