Solving Polynomial, and Exponential Equations and Inequalities

1
Solving Polynomial, and Exponential Equations and
Inequalities

• Mary Dwyer Wolfe, Ph.D.
• Department of Mathematics and Computer Science
• Macon State College
• MSP with Bibb County July 2010

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Math III Standard MM3A3

• Students will solve a variety of equations and
  inequalities.
• a. Find real and complex roots of higher degree
  polynomial equations using the factor theorem,
  remainder theorem, rational root theorem, and
  fundamental theorem of algebra, incorporating
  complex and radical conjugates.
• b. Solve polynomial, exponential, and logarithmic
  equations analytically, graphically, and using
  appropriate technology.
• c. Solve polynomial, exponential, and logarithmic
  inequalities analytically, graphically, and using
  appropriate technology. Represent solution sets
  of inequalities using interval notation.
• d. Solve a variety of types of equations by
  appropriate means choosing among mental
  calculation, pencil and paper, or appropriate
  technology.

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Focus of this Presentation

• b. Solve polynomial, exponential, and logarithmic
  equations analytically, graphically, and using
  appropriate technology.
• c. Solve polynomial, exponential, and logarithmic
  inequalities analytically, graphically, and using
  appropriate technology. Represent solution sets
  of inequalities using interval notation.

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Polynomial Equations
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Polynomial Equations

• Solve
• Steps in the Analytic Solution Process
• Put the equation in standard form
• Factor the polynomial into linear and quadratic
  terms
• Use the zero-product property

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Polynomial Equations

• Solve analytically (symbolically).

Standard form
Factor by grouping
Factor difference of squares
zero product property
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Polynomial Equations

• Solve graphically and numerically.

Standard form
graph
Solve by finding x so that f(x)0
Calculator Tutorial
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Polynomial Equations

• Solve graphically and numerically.

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Polynomial Equations

• Solve analytically (symbolically).

Standard form
Factor by grouping
zero product property
So if we can factor into linear and quadratic
factors, we can find the exact values of all real
and complex roots.
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Polynomial Equations

• Solve graphically and numerically.

Standard form
graph
Solve by finding x so that f(x)0
The calculator/graphing method can only find real
roots.
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Polynomial Equations

• Solve graphically and numerically.

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Polynomial Equations

• How do we find analytic solutions when the
  polynomial in standard form doesn't factor
  easily?
• Use the following
• Rational Root Theorem
• Factor Theorem
• Remainder Theorem
• -----and lots of trial and error to maybe factor
  the beast!
• See activity p. 15 Link to Activity
• BUT graphing can help conjecture rational roots
  along with the Rational Root Theorem!

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Polynomial Equations

• Real roots of polynomial equations can be
  approximated using numerical methods on the
  TI83/84 calculator. (The same numerical methods
  that I had to learn to do by HAND in the '60's!)
• Solve

There is one REAL root. (Rational Root Theorem
does not apply because the coefficients are not
integer.)
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Polynomial Equations

• Solve
• Graphical solution

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Polynomial Equations

• Solve
• by Intersection of Graphs Method.

• Method steps
• Enter left side of equation in Y1 and right side
  in Y2.
• Graph in a window where the intersection of the
  two functions is visible.
• Find the intersection point(s). The
  x-coordinates of these point(s) are the solutions.

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Polynomial Inequalities
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Polynomial Inequalities

• Solve analytically
• Solution
• Put in standard form
• Replace the gt with an and solve
• Note that these solutions to the equation are not
  solutions to the inequality

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Polynomial Inequalities

• Solve analytically
• Solution
• Put in standard form
• Put the solutions (x -2 or x 2) to the equation
  on a number line.
• Pick a test point in each interval formed and
  determine the sign of the inequality
• f(-4) -24 f(0) -8
  f(3) 25

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Polynomial Inequalities

• Put in standard form

f(x) lt 0 (negative) f(x) lt 0
(negative) f(x) gt 0 (positive) We are
looking for where f(x) gt 0 (is positive)
The solution is x gt 2 written in interval
notation is
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Polynomial Inequalities

• So you say that wasn't analytic enough, eh?

But other cubics could have up to 8 combinations
of 3 linear factors. This is an easy one!
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Polynomial Inequalities

• Put in standard form

The solution is x gt 2, or in interval notation

• We can get to the same conclusion using the
  TI83/84 calculator.
• Method
• Put the polynomial equation in standard form
• Enter the left side as a function
• Find the zeroes
• Examine the graph to determine position or
  negative f(x) values

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Polynomial Inequalities

• Put in standard form

Find the zeroes x-2 or x 2
Examine the graph to see that the positive value
occur when x gt 2
This is verified in the Table.
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Polynomial Inequalities

• Try this one
• Solve
• Solution

The solution set is
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Polynomial Inequality Application
The Chamber of Commerce in River City plans to
put on a 4th of July fireworks display. City
regulations require that the fireworks at public
gatherings explode higher than 800 feet from the
ground. The mayor particularly wants to include
the Freedom Starburst model, which is launched
from the ground. Its height after t seconds is
given by h 256t 16t2 When should the
Starburst explode in order to satisfy the safety
regulations? how many seconds to reach Height gt
800 feet? 256t 16t2 gt 800
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Polynomial Inequality Application
how many seconds to reach Height gt 800 feet? 256t
16t2 gt 800
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Polynomial Inequality Application
how many seconds to reach Height gt 800 feet? 256t
16t2 gt 800
4.3 11.7
The fireworks will be at a height of 800 feet or
more between 4.3 and 11.7 seconds after being
launched.
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Exponential Equations
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Exponential Equations

• Analytic (Symbolic) Method
• Solve for the exponential term and factor
• Take the log of both sides
• Use log of a power rule to get the variable out
  of the exponent
• Solve the resulting equation

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Exponential Equations

• Solve 2x 8 (Head problem!)
• x 3

Solve 2x 7 log 2x log 7 x (log 2) log 7 x
(log 7)/(log 2) x 2.807
Since the solution is irrational, the best we
get, even using symbolic methods, is a decimal
approximation.
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Exponential Equations
Solve 5(1.2)3x 2 95 100
5(1.2)3x 2 5 (1.2)3x 2 5/5
log (1.2)3x 2 log (1)
(3x 2) log (1.2) log (1)

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Exponential Equations

• An Application Newton's Law of Cooling

The temperature, T, of an object after time t is
modeled by T(t) T0 Dat where 0 lt a lt 1
and D is the initial temperature difference
between the object and the room. T0 is the
initial temperature of the environment.
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Exponential Equations

• An Application Newton's Law of Cooling

• Modeling Coffee Cooling
• A pot of coffee with temperature of 100C is
  placed in a room with a temperature of 20C. It
  takes one hour for the coffee to cool to 60C.
• Find the values of T0, D, and a for the formula
• T(t) T0 Dat
• (b) Find the temperature of the coffee after half
  and hour.
• (c) How long did it take for the coffee to reach
  50C?

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Exponential Equations

• A pot of coffee with temperature of 100C is
  placed in a room with a temperature of 20C. It
  takes one hour for the coffee to cool to 60C.
• Find the values of T0, D, and a for the formula
• T(t) T0 Dat
• T0 is the initial temperature of the room, so T0
  20
• D is the initial temperature difference D 100
  20 80
• So far we have T(t) 20 80at
• The last sentence tells us that T(1) 60
• So 60 20 80a1, so we can solve for a.
• 40 80a
• a 0.5
• so T(t) 20 80(0.5)t

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Exponential Equations
A pot of coffee with temperature of 100C is
placed in a room with a temperature of 20C. It
takes one hour for the coffee to cool to
60C. (a) T(t) 20 80(0.5)t (b) Find the
temperature of the coffee after half and
hour. That means find T(0.5) 20 80(0.5)0.5
76.6 So after half an hour the temperature of
the coffee is about 76.6C.
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Exponential Equations

• A pot of coffee with temperature of 100C is
  placed in a room with a temperature of 20C. It
  takes one hour for the coffee to cool to 60C.
• T(t) 20 80(0.5)t
• How long did it take for the coffee to reach
  50C?
• Solve 50 20 80(0.5)t

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Exponential Equations
Solve 50 20 80(0.5)t
t 1.415
Note that in this case, we went to a LOT of work
symbolically to arrive at only an approximate
result because the solution is an ugly irrational
number!
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Exponential Equations
Solve by Intersection of Graphs Method 50 20
80(0.5)t
t 1.415
Returning to our problem, it will take
approximately 1.415 hours for the coffee to cool
to 50C.
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Exponential Inequality
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Exponential Inequality

• I have 10,000 to invest. The current interest
  rate at my bank is 3.5 compounded daily. I want
  to leave the money in the account until it grows
  to somewhere between 20,000 and 30,000. How
  long must the money be left in the bank to grow
  to this range of amounts?
• So we must solve

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Exponential Inequality

• So we must solve

Rounded to the nearest year it will take between
20 and 32 years to have an amount between 20,000
and 30,000.
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Logarithmic Equations
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Logarithmic Equations

• Analytic solution method
• Rewrite equation as a single logarithmic
  statement
• Translate to an exponential statement
• Solve the resulting equation
• Solve ln 4x 1.5
• e1.5 4x

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Logarithmic Equations

• Solve log (x 1) log (x 1) log 3
• log (x 1)(x 1) log 3
• log (x 1)(x 1) - log 3 0

x -2 is extraneous, so the solution is x 2
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Logarithmic Equations

• Solve log (x 1) log (x 1) log 3
  Graphically

Pick a better window!
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Logarithmic Equations

• What about other log bases? Solve
• log2 x 3
• Where is the log2 key or menu item????

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Logarithmic Equations

• What about other log bases as in Solve
• log2 x 3
• Use the change of base formula

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Logarithmic Equations

• What about other log bases as in Solve
• log2 x 3
• Use the change of base formula

x 8
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Logarithmic Equations

• What about other log bases as in Solve
• log2 x 3
• Use the change of base formula

x 8
Note one problem with certain windows and log
functions
Press arrows to find a point on the first curve!
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Logarithmic Equations

• Try this one using the calculator method.
• Solve
• log3(x 24) log3(x 2) 2

x 0.75