Exponents

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Exponents Polynomials

• 6.1 Adding Subtracting Polynomials
• 6.2 Multiplying Polynomials
• 6.3 Multiplying Binomials Special Binomial
  Products
• 6.4 Problem Solving
• 6.5 Integral Exponents Dividing Polynomials
• 6.6 Dividing Polynomials by Binomials
• 6.7 Exponents Scientific Notation

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Polynomial Lexicon
A polynomial is defined as a single term or a
sum of two or more terms containing whole number
exponents on its variables.
- 9 x6 5x4 7x2 3x0
N.B. In standard form, polynomial terms are
written in order of descending powers.
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Polynomials Functions
Consider the equation of motion for a ball tossed
straight up into the air y -½ g t 2 vo t
yo where y vertical position of the tossed
ball in feet (ft) t time in seconds
(sec) g acceleration due to gravity 32
ft/sec2 vo initial ball speed in ft/sec yo
initial ball position (i.e. at t
0) Suppose a. that the vertical position of a
tossed fuchsia ball is described by the function
f(t) so that y f(t). where f(t) -16 t 2
60 t 50 b. that the vertical position of a
tossed green ball is described by the function
g(t) so that y g(t). where g(t) -16 t 2
75 t 50
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Polynomials FunctionsSymbolic Representation
a. the vertical position of a tossed fuchsia ball
at the times 0, 1, and 3 seconds is described by
the function f(t) so that f(t) -16 t 2
60 t 50 f(0) -16(0)2 60 (0) 50
50 ft f(1) -16(1)2 60 (1) 50 94
ft f(2) -16 (2)2 60 (2) 50 106
ft b. the vertical position of a tossed green
ball at the times 0, 1, and 3 seconds is
described by the function g(t) so that g(t)
-16 t 2 75 t 50 g(0) -16(0)2 75
(0) 50 50 ft g(1) -16(1)2 75 (1)
50 109 ft g(2) -16 (2)2 75 (2) 50
136 ft
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Polynomials FunctionsGraphical Numerical
Representations
f(t) -16t2 60 t 50 g(t) -16t2 75 t 50
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Adding Subtracting Polynomials
Subtracting Polynomials g(t) - f(t) (-16t 2
75 t 50 )- (-16t 2 60 t 50)
Adding Polynomials g(t) f(t) (-16t 2 75 t
50 ) (-16t 2 60 t 50)
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Adding Subtracting PolynomialsText Example
Given f(x) -9x3 7x2 -5x 3 g(x) 13x3
2x2 -8x -6
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Multiplying Polynomials
Case 1 Multiplying Monomials Product Rule for
Exponents If x is any real number and m and n are
natural numbers, then xm xn x mn Power Rule
for Exponential Expressions (xm)n x mn Power
Rule for Powers of Products (xy)m xmym
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Multiplying Polynomials
Case 2 Multiplying a Monomial Polynomial (Not
a Monomial) Distributive Property - Direct
Application e.g. (x3 2x2 - 4x 3) (-2x) Case
3 Products of Two Polynomials, Neither a
Monomial Distributive Commutative
Properties e.g. Horizontal Format (2x2 - 4x
3) (-2x 1) Vertical Format
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Multiplying Polynomials Special Binomial Products
( 3x 2 ) ( 4 x 5 ) Þ First terms F Þ (3x)
(4x) 12 x2 ( 3x 2 ) ( 4 x 5 ) Þ Outer
terms O Þ (3x) (5) 15 x ( 3x 2 ) ( 4 x 5 )
Þ Inner terms I Þ (2) (4x) 8 x ( 3x 2 ) ( 4
x 5 ) Þ Last terms L Þ (2) (5) 10
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More Special Products
The Square of a Binomial Sum (x y)2 x 2
2xy y 2 The Square of a Binomial Difference (x
- y)2 x 2 - 2xy y 2 Product of a Sum
Difference of Two Terms (x y) (x - y) x 2 -
y 2 FOIL - General Form (a b) (c d) ac ad
bc bd
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Area of a Square with Side Length (x y)
(x y )2 (x y ) (x y ) F O I L
A1 A2 A3 A4 x2 xy xy y2 x2
2xy y2
A1 x2
A4 xy
A2 xy
A3 y2
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Problem Solving
The square painting shown is surrounded by a
frame that uniformly measures 1 inch wide. If
the frame area is 28 in2, find the dimensions of
the painting.
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Integral Exponents
Quotient Rule for Exponents If x is any nonzero
real number and m and n are natural numbers,
then Definition of Zero Exponent If x is any
nonzero real number, then x0 1 Definition of a
Negative Integer as an Exponent If x is any
nonzero real number and n is any integer,
then If x ¹ 0, then
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Dividing Polynomials by Binomials
Long Division of Polynomials 1. Arrange the terms
of both the dividend and the divisor in
descending powers of the variable 2. Divide the
first term in the dividend by the first term in
the divisor. The result will be the first term
in the quotient. 3. Multiply every term in the
divisor by the first term in the quotient. Write
the resulting product beneath the dividend with
similar terms under each other. 4. Subtract the
product from the dividend. 5. Bring down the next
term in the original dividend and write it next
to the remainder to form a new dividend. 6. Use
this new expression as the dividend and repeat
this process until the degree of the remainder is
smaller than the degree of the divisor.
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Dividing Polynomials by Binomials - Examples
Divide 3983 by 26 3x3 9x2 8x 3 by 2x2 6
x2 10x 21 by x 3 7x - 9 - 4x2 4x3 by 2x
- 1
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Exponents Scientific Notation
The mass of a jumbo jet is about 375,000
kilograms. Scientific Notation (Report
Format) 3.75 x 105 kg Scientific Notation
(Calculator Format) 3.75 E05 kg The radius of
an atom is about 1/10000000000 metres Scientific
Notation (Report Format) 1.0 x 10-10
m Scientific Notation (Calculator Format) 1.0
E-10 m
3 7 5 0 0 0
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Integral Exponents Definitions Properties
Definitions 1. x1 x 2. xn means x repeated n
times as a factor, e.g., x4 x . x . x .
x 3. x0 1, if x ¹ 0 4. if x ¹
0 Properties 1. xm xn x mn 2. (xm)n x
mn 3. (xy)m xmym 4. 5.
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MTH_065 Chapter 6 Exam

• Instructions and Remarks
• Read all of the following instructions before
  starting work on this examination.
• This examination consists of four problems on two
  pages.
• A calculator and one 8½ x 11 cheat sheet is
  allowed during this examination.
• Examination solutions shall be submitted on a
  separate sheet(s) of paper.
• Ensure that your name, todays date, and the
  title MTH_065 Exam I appears on top of your
  examination solution sheet.
• Ensure that your solution and the associated
  problem number are clearly identified.
• Note that marks carried by each portion of the
  examination are indicated in square brackets
  (i.e., ).
• Enjoy!