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n −1 Theorem 1: If integer roots exist for an equation x + an −1 x + ... + a1 x + a0
Theorem
integer
with integer coefficients, then each root must be a divisor of a0.
with
n =0 Indeed, in the above example, the set of potential integer roots is
{1,1,2,2,4,4}.
All of the actual rational roots belong to this set.
14 Rational Roots
A useful theorem helps us search for the rational (not just integer!) roots
useful
of a general polynomial equation.
of
n −1 Theorem 2: If rational roots exist for an equation an x + an −1 x + ... + a1 x + a0 = 0
Theorem
rational
with integer coefficients then the nominator of a rational root is a divisor of
with
an
a0 hile the denominator of this rational root is a divisor of
,w
n Example 2 x 4 + 5 x 3 −11x 2 − 20 x + 12 = 0 Step 1. All possible nominators of the potential rational roots have to
Step
belong to the set of the divisors of a0 =12, which is R={1,1,2,,2,3,3,4,belong
4,6,6,12,12}
Step 2. All possible denominators of the potential rational roots have to
Step
belong to the set of the divisors of an = 2 , which is S={1,1,2,2}
belong
Notice that potential divisors include unity and the coefficient itself!
Step 3. All possible ratios of elements in R to elements in S will contain
Step
all of the quatric equation’s rational roots, which are: ½,2,2,3.
15 Tips to Solve Polynomial Equations
1) Stop looking for the additional rational roots once you have found
Stop
the maximum possible number of the rational roots, given by the
maximum
degree of the underlying polynomial.
degree
2) When an = 1 , the set containing possible rational roots is simply the
When
possible
set of all of the divisors of a0 .
set
3) If the coefficients of a polynomial equation are not integer, you can
If
always transform it by multiplying both sides of the equation by the
maximum value of the coefficients, max{ a0 , a1 ,..., an }
i = 0..n 16 Market Equilibrium for Two Goods
In general, quantity demanded and supplied for a good depend not only
In
on its own price, but also on the price of the related goods:
but Qd 1 = a0 + a1 P1 + a2 P2
Read as: demand for good 1 depends on its own price P AND on
Read
1
price P of a related good.
price 2
Economically sensible constraints on the parameters:
1) a1 < 0 : if we believe that consumers dislike expensive things
if
2) a2 < >0 : Since the second good can be a substitute or a complement
Since
1) If the second good is a complement, a2 < 0
2) If the second good is a substitute, a2 > 0 17 Market Equilibrium for Two Goods
Formally, a model of general equilibrium involving two related goods will
Formally,
be written down as:
be
Q
Q
0 d 1 − s1 =
Equilibrium for good Q
a
a1
a
Demand for good 1
2 d1 = 0 + 1P + 2 P s1 = 0 + 1 P + 2 P
Q
b
b1
b2
Supply for good 1 Q
Q
0
Equilibrium for good d 2 − s2 = d 2 = 0 + 1 P + 2 P Demand for good 2
Q
α α1 α 2 s 2 = 0 + 1 P + 2 P Supply for good 2
Q
β β1 β 2 1
2 By eliminating the variables (i.e. by making use of the two eq...
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 Fall '11
 Kim
 Equilibrium, Supply And Demand, Qd

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