Chapter 2: Polynomials
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)
`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.
Bounding roots: start with an > 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.
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)
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.
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
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
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
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
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]
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